J-holomorphic Curves and Quantum Cohomology

Transcription

J-holomorphic Curves
and Quantum Cohomology
by
Dusa McDuff and Dietmar Salamon
May 1995
Contents
1 Introduction
1.1 Symplectic manifolds . . . . . . . .
1.2 J-holomorphic curves . . . . . . .
1.3 Moduli spaces . . . . . . . . . . . .
1.4 Compactness . . . . . . . . . . . .
1.5 Evaluation maps . . . . . . . . . .
1.6 The Gromov-Witten invariants . .
1.7 Quantum cohomology . . . . . . .
1.8 Novikov rings and Floer homology
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1
1
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2 Local Behaviour
13
2.1 The generalised Cauchy-Riemann equation . . . . . . . . . . . . . . . 13
2.2 Critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Somewhere injective curves . . . . . . . . . . . . . . . . . . . . . . . 18
3 Moduli Spaces and Transversality
3.1 The main theorems . . . . . . . .
3.2 Elliptic regularity . . . . . . . . .
3.3 Implicit function theorem . . . .
3.4 Transversality . . . . . . . . . . .
3.5 A regularity criterion . . . . . . .
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23
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4 Compactness
4.1 Energy . . . . . . . . . . . . .
4.2 Removal of Singularities . . .
4.3 Bubbling . . . . . . . . . . .
4.4 Gromov compactness . . . . .
4.5 Proof of Gromov compactness
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41
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5 Compactification of Moduli Spaces
5.1 Semi-positivity . . . . . . . . . . . . . .
5.2 The image of the evaluation map . . . .
5.3 The image of the p-fold evaluation map
5.4 The evaluation map for marked curves .
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59
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vii
viii
CONTENTS
6 Evaluation Maps and Transversality
6.1 Evaluation maps are submersions . . . . .
6.2 Moduli spaces of N -tuples of curves . . .
6.3 Moduli spaces of cusp-curves . . . . . . .
6.4 Evaluation maps for cusp-curves . . . . .
6.5 Proofs of the theorems in Sections 5.2 and
6.6 Proof of the theorem in Section 5.4 . . . .
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5.3
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7 Gromov-Witten Invariants
7.1 Pseudo-cycles . . . . . . .
7.2 The invariant Φ . . . . . .
7.3 Examples . . . . . . . . .
7.4 The invariant Ψ . . . . . .
71
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81
82
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89
. 90
. 93
. 98
. 101
8 Quantum Cohomology
8.1 Witten’s deformed cohomology ring
8.2 Associativity and composition rules .
8.3 Flag manifolds . . . . . . . . . . . .
8.4 Grassmannians . . . . . . . . . . . .
8.5 The Gromov-Witten potential . . . .
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107
107
114
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123
130
Manifolds
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148
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9 Novikov Rings and Calabi-Yau
9.1 Multiply-covered curves . . .
9.2 Novikov rings . . . . . . . . .
9.3 Calabi-Yau manifolds . . . .
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10 Floer Homology
10.1 Floer’s cochain complex . . . .
10.2 Ring structure . . . . . . . . .
10.3 A comparison theorem . . . . .
10.4 Donaldson’s quantum category
10.5 Closing remark . . . . . . . . .
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153
153
159
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162
165
A Gluing
A.1 Cutoff functions . . . . . . . . . . . . . . .
A.2 Connected sums of J-holomorphic curves
A.3 Weighted norms . . . . . . . . . . . . . .
A.4 An estimate for the inverse . . . . . . . .
A.5 Gluing . . . . . . . . . . . . . . . . . . . .
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167
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176
B Elliptic Regularity
B.1 Sobolev spaces . . . . . . . . . . .
B.2 The Calderon-Zygmund inequality
B.3 Cauchy-Riemann operators . . . .
B.4 Elliptic bootstrapping . . . . . . .
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181
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185
190
192
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Preface
The theory of J-holomorphic curves has been of great importance to symplectic
topologists ever since its inception in Gromov’s paper [26] of 1985. Its applications include many key results in symplectic topology: see, for example, Gromov [26], McDuff [42, 45], Lalonde–McDuff [36], and the collection of articles in
Audin–Lafontaine [5]. It was also one of the main inspirations for the creation
of Floer homology [18, 19, 73], and recently has caught the attention of mathematical physicists through the theory of quantum cohomology: see Vafa [82] and
Aspinwall–Morrison [2].
Because of this increased interest on the part of the wider mathematical community, it is a good time to write an expository account of the field, which explains the
main technical steps in the theory. Although all the details are available in the literature in some form or other, they are rather scattered. Also, some improvements in
exposition are now possible. Our account is not, of course, complete, but it is written with a fair amount of analytic detail, and should serve as a useful introduction
to the subject. We develop the theory of the Gromov-Witten invariants as formulated by Ruan in [64] and give a detailed account of their applications to quantum
cohomology. In particular, we give a new proof of Ruan–Tian’s theorem [67, 68]
that the quantum cup-product is associative.
Many people have made useful comments which have added significantly to our
understanding. In particular, we wish to thank Givental for explaining quantum
cohomology, Ruan for several useful discussions and for pointing out to us the connection between associativity of quantum multiplication and the WDVV-equation,
Taubes for his elegant contribution to Section 3.4, and especially Gang Liu for
pointing out a significant gap in an earlier version of the gluing argument. We
are also grateful to Lalonde for making helpful comments on a first draft of this
manuscript. The first author wishes to acknowledge the hospitality of the University of California at Berkeley, and the grant GER-9350075 under the NSF Visiting
Professorship for Women program which provided partial support during some of
the work on this book.
Dusa McDuff and Dietmar Salamon,
June 1994.
ix
x
CONTENTS
2007 Commentary on J-holomorphic curves and Quantum Cohomology
We have decided to make our first set of lecture notes on J-holomorphic curves
available on the Web since, despite having a few mistakes and many omissions, it is
still a readable introduction to the field. It was originally published by the AMS in
1994. The version here is the revised edition first published in May 1995. It is now
out of print, replaced by the enlarged version J-holomorphic curves and Symplectic
Topology (AMS 2004) that we will refer to as JHOL.
Main mistakes and omissions
1. The proof the Calderon–Zygmund inequality in Appendix B.2 was simply
wrong. A correct proof is given in JHOL Appendix B.
2. The proof of Gromov compactness was short, but somewhat careless. Claim
(ii) on p 56 in the proof of the Gromov compactness theorem 4.4.3 is wrong; the
limiting curve v might be constant. We used the claim that v is nonconstant to
show that the limiting process converges after finitely many steps.
To correct this one needs to take more care in the rescaling argument, namely the
2
center of the rescaling must be chosen at a point where the function |duν | attains
a local maximum (Step 1 in the proof of Proposition 4.7.1, p. 101 in JHOL). In this
way one obtains two additional bubbling points at the center and on the boundary
of the unit disc for the rescaled sequence, when the limit is constant (see (iv) in
Proposition 4.7.1 and Step 4 in the proof, p. 104 in JHOL).
The refined rescaling argument gives as a limit a stable map, a crucial missing
ingredient in the old version. One then uses JHOL Exercise 5.1.2 to establish a
bound on the number of components of the limiting stable map.
3. An omission in the Gluing proof in Appendix A. The proof of the existence
of the gluing map fR of Theorem A.5.2 is basically all right, but somewhat sketchy.
In particular we did not prove in detail that it is a local diffeomorphism. We claim
in the middle of p. 175 that this follows from the uniqueness result in Proposition
3.3.5. While this is true, several intermediate steps are needed for a complete proof.
Full details are given in JHOL Chapter 10.
4. An omission in the transversality argument. In Chapter 2 we established
basic results on the structure of simple J-holomorphic curves only in the case of
C ∞ -smooth J. However, in the application in Chapter 3 we need these results
for J of class C ` with ` < ∞. This was somewhat concealed: the statement of
Proposition 3.4.1 does not make clear that the elements in the universal moduli
space M` (A, J) are assumed to be simple although we used that assumption in the
proof.
Additions in JHOL
Many of the discussions in this book are quite sketchy. In JHOL they are all fleshed
out, and there are many more examples given. Here are other main additions:
CONTENTS
xi
• The existence of Gromov–Witten invariants is established in more generality
(though the argument still needs some version of the “semipositive” hypothesis).
Chapter 7 discusses the axioms they satisfy and also has a completely self-contained
section on the very special and important example of rational curves in projective
space;
• a treatment of genus zero stable curves and maps (in Appendix D and Chapter
5);
• a chapter on geometric applications of genus zero J-holomorphic curves without
using gluing (Chapter 9);
• a discussion of Hamiltonian Floer homology and applications such as spectral invariants, explaining the set up but without doing the basic analytic proofs (Chapter
12);
• much more detailed appendices on the analysis. Also an appendix giving full
details of a new proof (due to Lazzarini) on positivity of intersections.
Dusa McDuff and Dietmar Salamon, October 2007.
xii
CONTENTS
Chapter 1
Introduction
The theory of J-holomorphic curves is one of the new techniques which have recently
revolutionized the study of symplectic geometry, making it possible to study the
global structure of symplectic manifolds. The methods are also of interest in the
study of Kähler manifolds, since often when one studies properties of holomorphic
curves in such manifolds it is necessary to perturb the complex structure to be
generic. The effect of this is to ensure that one is looking at persistent rather than
accidental features of these curves. However, the perturbed structure may no longer
be integrable, and so again one is led to the study of curves which are holomorphic
with respect to some non-integrable almost complex structure J.
The present book has two aims. The first is to establish the fundamental theorems in the subject, and in particular the existence of the Gromov-Witten invariants. The second is to serve as an introduction to the subject. These two aims are,
of course, somewhat in conflict, and in different parts of the book different aspects
are predominant. The book is written in logical order. Chapters 2 through 6 establish the foundational Fredholm theory and compactness results needed to set up the
theory. Chapters 7 and 8 then discuss the Gromov-Witten invariants, and their application to quantum cohomology, for symplectic manifolds which satisfy a certain
positivity (or monotonicity) condition. Chapter 9 treats extensions to Calabi-Yau
manifolds, and Chapter 10 relations with Floer homology. Appendix A develops
a gluing technique for for J-holomorphic spheres, which is an essential ingredient
of the proof of the associativity of quantum multiplication. Finally, Appendix B
reviews and proves some basic facts on elliptic regularity.
It may be best not to read the book in chronological order, but rather to start
with Chapter 7. The present introductory chapter aims to outline enough of the
theory of J-holomorphic curves to make this approach feasible. We assume that
the reader is familiar with the elements of symplectic geometry. Good references
are McDuff–Salamon [52] and the introductory articles in Audin–Lafontaine [5].
1.1
Symplectic manifolds
A symplectic structure on a smooth 2n-dimensional manifold M is a closed 2-form
ω which is non-degenerate in the sense that the top-dimensional form ω n does not
1
2
CHAPTER 1. INTRODUCTION
vanish. By Darboux’s theorem, all symplectic forms are locally diffeomorphic to
the standard linear form
ω0 = dx1 ∧ dx2 + · · · + dx2n−1 ∧ dx2n
on Euclidean space R2n . This makes it hard to get a handle on the global structure
of symplectic manifolds. Variational techniques have been developed which allow
one to investigate some global questions in Euclidean space and in manifolds such
as cotangent bundles which have some linear structure: see [52] and the references
contained therein. But the method which applies to the widest variety of symplectic
manifolds is that of J-holomorphic curves.
Here J is an almost complex structure on M which is tamed by ω. An almost
complex structure is an automorphism J of the tangent bundle T M of M which
satisfies the identity J 2 = −1l. Thus J can be thought of as multiplication by i,
and it makes T M into a complex vector bundle of dimension n. The form ω is said
to tame J if
ω(v, Jv) > 0
for all nonzero v ∈ T M . Geometrically, this means that ω restricts to a positive
form on each complex line L = span{v, Jv} in the tangent space Tx M . Given ω
the set Jτ (M, ω) of almost complex structures tamed by ω is always non-empty
and contractible. Note that it is very easy to construct and perturb tame almost
complex structures, because they are defined by pointwise conditions. Note also
that, because Jτ (M, ω) is path-connected, different choices of J ∈ Jτ (M, ω) give
rise to isomorphic complex vector bundles (T M, J). Thus the Chern classes of these
bundles are independent of the choice of J and will be denoted by ci (M ).
In what follows we shall only need to use the first Chern class, and what will
be relevant is the value which it takes on embedded 2-spheres in M . If S is such
a sphere in the homology class A ∈ H2 (M ), we will need to calculate c1 (A). This
is just the first Chern class of the restriction of the bundle T M to S. But every
complex bundle E over a 2-sphere S decomposes as a sum of complex line bundles
L1 ⊕ · · · ⊕ Ln . Correspondingly
X
c1 (E) =
c1 (Li ).
i
Since the first Chern class of a complex line bundle is just the same as its Euler
class, it is often easy to calculate the c1 (Li ) directly. For example, if A is the class
of the sphere S = pt × S 2 in M = V × S 2 then it is easy to see that
T M S = T S ⊕ L2 ⊕ · · · ⊕ Ln ,
where the line bundles Lk are trivial. It follows that
c1 (A) = c1 (T M S ) = c1 (T S) = χ(S) = 2
where χ(S) is the Euler characteristic of S.
A smooth map φ : (M, J) → (M 0 , J 0 ) from one almost complex manifold to
another is said to be (J, J 0 )-holomorphic if and only if its derivative dφx : Tx M →
Tφ(x) M 0 is complex linear, that is
0
dφx ◦ Jx = Jφ(x)
◦ dφx .
1.2. J-HOLOMORPHIC CURVES
3
These are the Cauchy-Riemann equations, and, when (M, J) and (M 0 , J 0 ) are both
subsets of complex n-space Cn , they are satisfied exactly by the holomorphic maps.
An almost complex structure J is said to be integrable if it arises from an underlying complex structure on M . This is equivalent to saying that one can choose an
atlas for M whose coordinate charts are (J, i)-holomorphic where i is the standard
complex structure on Cn . In this case the coordinate changes are holomorphic maps
(in the usual sense) between open sets in Cn . When M has dimension 2 a fundamental theorem says that all almost complex structures J on M are integrable.
However this is far from true in higher dimensions.
The basic example of an almost complex symplectic manifold is standard Euclidean space (R2n , ω0 ) with its standard almost complex structure J0 obtained
from the usual identification with Cn . Thus, one sets
zj = x2j−1 + ix2j
for j = 1, . . . , n and defines J0 by
J0 (∂2j−1 ) = ∂2j ,
J0 (∂2j ) = −∂2j−1
where ∂j = ∂/∂xj is the standard basis of Tx R2n . Kähler manifolds give another
basic example.
1.2
J-holomorphic curves
A J-holomorphic curve is a (j, J)-holomorphic map
u:Σ→M
from a Riemann surface (Σ, j) to an almost complex manifold (M, J).1 Usually,
we will take (Σ, j) to be the Riemann sphere S 2 = C ∪ {∞}. In accordance with
the terminology of complex geometry it is often convenient to think of the 2-sphere
as the complex projective line CP 1 . If u is an embedding (that is, an injective
immersion) then its image C is a 2-dimensional submanifold of M whose tangent
spaces Tx C are J-invariant. Thus each tangent space is a complex line in T M .
Conversely, any 2-dimensional submanifold C of M with a J-invariant tangent
bundle T C has a J-holomorphic parametrization u. This follows immediately from
the fact that the restriction of J to C is integrable.
Note that according to this definition, a curve u is always parametrized. One
should contrast this with the situation in complex geometry, where one often defines
a curve by requiring it to be the common zero set of a certain number of holomorphic polynomials. Such an approach makes no sense in the non-integrable, almost
complex context, since when J is non-integrable there usually are no holomorphic
functions (M, J) → C.
By the taming condition, ω restricts to a positive form on each such line. Therefore C is a symplectic submanifold of M .2 Conversely, given a 2-dimensional symplectic submanifold of M , it is not hard to construct an ω-tame J such that T C
1A
2A
Riemann surface is a 1-dimensional complex manifold.
submanifold X of M is said to be symplectic if ω restricts to a non-degenerate form on X.
4
is J-invariant. (First define J on T C, then extend to the tangent spaces Tx M for
x ∈ C, and finally extend the section to the rest of M .) Thus J-holomorphic curves
are essentially the same as 2-dimensional symplectic submanifolds of M .
1.3
Moduli spaces
The crucial fact about J-holomorphic curves is that when J is generic they occur
in finite dimensional families. These families make up finite dimensional manifolds
M(A, J)
which are called moduli spaces and whose cobordism classes are independent of
the particular J chosen in Jτ (M, ω). Here A is a homology class in H2 (M, Z),
and M(A, J) consists of essentially all J-holomorphic curves u : Σ → M which
represent the class A. Although the manifold M(A, J) is almost never compact,
it usually retains enough elements of compactness for one to be able to use it to
define invariants.
Chapters 2-6 of this book are taken up with formulating and proving precise
versions of the above statements. Here is a brief description of the main results.
Local properties
The first chapter is concerned with local properties of J-holomorphic curves. The
key result for future developments is perhaps Proposition 2.3.1, which gives a characterization of those curves which are not multiply-covered. A curve u : Σ → M
is said to be multiply-covered if it is the composite of a holomorphic branched
covering map (Σ, j) → (Σ0 , j 0 ) of degree greater than 1 with a J-holomorphic map
Σ0 → M . It is called simple if it is not multiply-covered. The multiply-covered
curves are often singular points in the moduli space M(A, J), and so one needs a
workable criterion which guarantees that u is simple. We will say that a curve u is
somewhere injective if there is a point z ∈ Σ such that
du(z) 6= 0,
u−1 (u(z)) = {z}.
A point z ∈ Σ with this property is called an injective point for u.
Proposition 1.3.1 Every simple J-holomorphic curve u is somewhere injective.
Moreover the set of injective points is open and dense in Σ.
Fredholm theory
Fix a Riemann surface Σ of genus g and denote by M(A, J) the set of all simple
J-holomorphic maps u : Σ → M which represent the class A. In Chapter 3 we
prove the following theorem.
Theorem 1.3.2 There is a subset Jreg (A) ⊂ Jτ (M, ω) of the second category (i.e.
it contains a countable intersection of open and dense sets) such that for each
J ∈ Jreg (A) the space M(A, J) is a smooth manifold of dimension
dim M(A, J) = n(2 − 2g) + 2c1 (A).
This manifold M(A, J) carries a natural orientation.
1.4. COMPACTNESS
5
Another important theorem specifies the dependence of M(A, J) on the choice
of J (Theorem 3.1.3).
The basic reason why these theorems are valid is that the Cauchy-Riemann
equation
du ◦ j = J ◦ du
is elliptic, and hence its linearization is Fredholm. A bounded linear operator L
from one Banach space X to another Y is said to be Fredholm if it has a finite
dimensional kernel and a closed range L(X) of finite codimension in Y . The index
of L is defined to be the difference in dimension between the kernel and cokernel of
L:
index L = dim ker L − dim cokerL.
An important fact is that the set of Fredholm operators is open with respect to the
norm topology and the Fredholm index is constant on each component of the set
of Fredholm operators. Thus, it does not change as L varies continuously, though
of course the dimension of the kernel and cokernel can change.
Fredholm operators are essentially as well-behaved as finite-dimensional operators and they play an important role in infinite dimensional implicit function theorems. Thus, if F : X → Y is a a C ∞ -smooth map whose derivative dF(x) : X → Y
is Fredholm of index k at each point x ∈ X and if y ∈ Y is a regular value of
F in the sense that dF(x) is surjective for all x ∈ F −1 (y), then, just as in the
finite-dimensional case, the inverse image
F −1 (y)
is a smooth manifold of dimension k. An infinite dimensional version of Sard’s
theorem says that almost all points of Y are regular for F. Technically, they form a
set of second category. This theorem remains true if X and Y are Banach manifolds
rather than Banach spaces. However it does not extend as stated to other kinds of
infinite dimensional vector spaces, such as Fréchet spaces.
The set Jreg mentioned in the above theorem does consist of the regular values
of some Fredholm operator which maps into the space Jτ (M, ω). There are some
additional technicalities in the proof which are caused by the fact that Jτ (M, ω) is
a Fréchet rather than a Banach manifold. Elements J ∈ Jτ (M, ω) which belong to
the subset Jreg are often said to be generic. An interesting fact is that the taming
condition on J is irrelevant here. The above results hold for all almost complex
structures J on any compact manifold M .
1.4
Compactness
The next task is to develop an understanding of when the moduli spaces M(A, J)
are compact. Here the taming condition plays an essential role. The symplectic
form ω and an ω-tame almost complex structure J together determine a Riemannian
metric
hv, wi = 21 (ω(v, Jw) + ω(w, Jv))
6
on M and the energy of a J-holomorphic curve u : Σ → M with respect to this
metric is given by the formula
Z
Z
E(u) =
|du|2 =
u∗ ω.
Σ
Σ
Thus the L2 -norm of the derivative of a J-holomorphic curve satisfies a uniform
bound which depends only on the homology class A represented by u. This in
itself does not guarantee compactness because it is a borderline case for Sobolev
estimates. (A uniform bound on the Lp -norms of du with p > 2 would guarantee
compactness.)
Another manifestation of the failure of compactness can be observed in the fact
that the energy E(u) is invariant under conformal rescaling of the metric on Σ.
This effect is particularly clear in the case where the domain Σ of our curves is the
Riemann sphere CP 1 , since here there is a large group of global, rather than local,
rescalings. Indeed, the non-compact group G = PSL(2, C) acts on the Riemann
sphere by conformal transformations
az + b
.
cz + d
Thus each element u ∈ M(A, J) has a non-compact family of reparametrizations
u ◦ φ, for φ ∈ G, and so M(A, J) itself can never be both compact and non-empty
(unless A is the zero class, in which case all the maps u are constant). However,
the quotient space C(A, J) = M(A, J)/G will sometimes be compact.
Recall that a homology class B ∈ H2 (M ) is called spherical if it is in the image
of the Hurewicz homomorphism π2 (M ) → H2 (M ). Here is a statement of the first
important theorem in Chapter 4.
z 7→
Theorem 1.4.1 Assume that there is no spherical homology class B ∈ H2 (M )
such that 0 < ω(B) < ω(A). Then the moduli space M(A, J)/G is compact.
One proves this by showing that, if uν is a sequence in M(A, J) which has no
limit point in M(A, J), then there is a point z ∈ CP 1 at which the derivatives
duν (z) are unbounded. This implies that, after passage to a subsequence, there
is a decreasing sequence Uν of neighbourhoods of z in CP 1 whose images uν (Uν )
converge in the limit to a J-holomorphic sphere. If B is the homology class represented by this sphere, then either ω(B) = ω(A), in which case the maps uν can be
reparametrized so that they do converge in M(A, J), or ω(B) lies strictly between
0 and ω(A). This is the process of “bubbling”, which occurs in this context in a
simple and geometrically clear way.
One consequence of this theorem is that if ω(A) is already the smallest positive
value assumed by ω on spheres then the moduli space M(A, J)/G is compact. To
go further, we need a sharper form of this result, which describes the complete limit
of the sequence uν . This is Gromov’s compactness Theorem 4.4.3, which is stated
in Section 4.4. A complete proof is given in Section 4.5.
1.5
Evaluation maps
The invariants which we shall use are built from the evaluation map
M(A, J) × CP 1 → M : (u, z) 7→ u(z).
1.5. EVALUATION MAPS
7
Note that this factors through the action of the reparametrization group G given
by
φ · (u, z) = (u ◦ φ−1 , φ(z)).
Hence we get a map defined on the quotient
e = eJ : W(A, J) = M(A, J) ×G CP 1 → M.
Example 1.5.1 Suppose that M is a product CP 1 × V with a product symplectic
form and let A = [CP 1 × pt]. Suppose also that π2 (V ) = 0. Then A generates
the group of spherical 2-classes in M , and so ω(A) is necessarily the smallest value
assumed by ω on the spherical classes. Theorems 3.1.2 and 4.3.2 therefore imply
that the space W(A, J) is a compact manifold for generic J. Because c1 (A) = 2,
in this case the dimension of W(A, J) is 2n and agrees with the dimension of M .
Moreover, we will see in Chapter 3 (see Theorem 3.1.3) that different choices of J
give rise to cobordant maps eJ . Since cobordant maps have the same degree, this
means that the degree of eJ is independent of all choices. Now if J = i × J 0 is a
product, where i denotes the standard complex structure on CP 1 , then it is easy
to see that the elements of M(A, J) have the form
u(z) = (φ(z), v0 )
where v0 ∈ V and φ ∈ G. It follows that the map eJ has degree 1 for this choice of
J and hence for every J.
Gromov used this fact in [26] to prove his celebrated non-squeezing theorem.
Theorem 1.5.2 If ψ is a symplectic embedding of the ball B 2n (r) of radius r into
a cylinder B 2 (λ) × V , where π2 (V ) = 0, then r ≤ λ.
Sketch of proof: Embed the disc B 2 (λ) into a 2-sphere CP 1 of area πλ2 + ε, and
let ω be the product symplectic structure on CP 1 × V . Let J 0 be an ω-tame almost
complex structure on CP 1 × V which, on the image of ψ, equals the push-forward
by ψ of the standard structure J0 of the ball B 2n (r). Since the evaluation map
eJ 0 has degree 1, there is a J 0 -holomorphic curve through every point of CP 1 × V .
In particular, there is such a curve, C 0 say, through the image ψ(0) of the center
of the ball. This curve pulls back by ψ to a J0 -holomorphic curve C through the
center of the ball B 2n (r). Since J0 is standard, this curve C is holomorphic in the
usual sense and so is a minimal surface in B 2n (r). But it is a standard result in the
theory of minimal surfaces, that the surface of smallest area which goes through
the center of a ball in Euclidean space is the flat disc of area πr2 . Further, because
C is holomorphic, it is easy to check that its area is just given by the integral of
the standard symplectic form ω0 over it. Thus
Z
Z
Z
πr2 ≤
ω0 =
ψ ∗ (ω) <
ω = ω(A) = πλ2 + ε
C
ψ −1 (C 0 )
C0
where the middle inequality holds because ψ(C) is only a part of C 0 . Since this is
true for all ε > 0, the result follows. More details may be found in [26, 43, 36, 52].
2
8
Often it is useful to evaluate the map u at more than one point and so we shall
consider the maps
ep : W(A, J, p) = M(A, J) ×G (CP 1 )p → M p
defined by
ep (u, z1 , . . . , zp ) = (u(z1 ), . . . , u(zp )).
Here p is any positive integer and M p = M ×· · ·×M . For a generic almost complex
structure the domain W(A, J, p) of this evaluation map is a manifold of dimension
dim W(A, J, p) = 2n + 2c1 (A) + 2p − 6.
In general this manifold will not be compact. However, in many cases one can show
that its image X(A, J, p) = ep (W(A, J, p)) ⊂ M p can be compactified by adding
pieces of dimension at least 2 less than that of W(A, J, p). This is the content
of the following theorem. For the purposes of exposition, we state it in a slightly
simplified form. In particular, we assume that (M, ω) is monotone. This means
that there is a positive constant λ > 0 such that
ω(B) = λc1 (B)
for all spherical classes B ∈ H2 (M ; Z), where c1 is the first Chern class of the
complex bundle (T M, J). This hypothesis is stronger than necessary, but has the
virtue of being easy to understand.
Theorem 1.5.3 Let (M, ω) be a monotone compact symplectic manifold and A ∈
H2 (M, Z).
(i) For every J ∈ Jτ (M, ω) there exists a finite collection of evaluation maps ek :
Wk (J) → M p such that
[
\
ep (W(A, J, p) − K) ⊂
ek (Wk (J)).
K⊂W(A,J,p)
K compact
k
(ii) There is a set of second category in Jτ (M, ω) such that, for every J in this set,
the spaces Wk (J) are smooth oriented σ-compact manifolds of dimensions
dim Wk (J) ≤ dim W(A, J, p) − 2.
A collection of such theorems is stated and discussed in Chapter 5. They are
proved in Chapter 6.
1.6
The Gromov-Witten invariants
The above theorem implies that the map ep : W(A, J, p) → M p represents a welldefined homology class in M p . Intuitively, its image is an m-chain, where m =
dim W(A, J, p), whose boundary has dimension at most m − 2 and so is not seen
from homological point of view. In Chapter 7 this is formalised in the notion
of a pseudo-cycle. One can show that the homology class represented by the
1.7. QUANTUM COHOMOLOGY
9
evaluation map ep does not depend on the choice of J. The Gromov invariants
are obtained by taking its intersection with cycles of complementary dimension in
M p.
More precisely, let α = α1 × · · · × αp be an element of Hd (M p ) where
d + dim W(A, J, p) = 2np.
Then we may choose a representing cycle for α – also denoted by α – which intersects
the image of ep : W(A, J, p) → M p transversely in a finite set of points. We define
the Gromov invariant
ΦA (α1 , . . . , αp ) = ep · α
to be the number of these intersection points counted with signs according to their
orientations. This is simply the number of J-holomorphic curves u in the homology
class A which meet each of the cycles α1 , . . . , αp . If the dimensional condition is
not satisfied, we simply set ΦA (α1 , . . . , αp ) = 0. Some examples are calculated in
Section 7.3. In particular, we sketch Ruan’s argument which uses these invariants
to show that there are non-deformation equivalent 6-manifolds. The discussion in
Sections 7.1 and 7.2 is slightly more technical than the present one mostly because
we want to weaken the hypothesis of monotonicity. However, the examples in
Section 7.3 should be accessible at this point.
The rest of Chapter 7 is devoted to a discussion of the Gromov-Witten invariant
Ψ. Here one fixes a p-tuple of distinct points z1 , . . . , zp in CP 1 , and counts the
number of curves u such that
u(z1 ) ∈ α1 , . . . , u(zp ) ∈ αp .
Again, the dimensions are chosen so that there will only be finitely many such
curves. Because the reparametrization group G is triply transitive, it is possible to
do this only if p ≥ 3. Moreover, in the case p = 3 both invariants agree
ΦA (α1 , α2 , α3 ) = ΨA (α1 , α2 , α3 ).
However, when p > 3 the two invariants are rather different. In particular, it turns
out that the invariants Ψ have much simpler formal properties. For example, we
shall see in Section 8.2 that they obey the decomposition rule
XX
ΨA (α1 , α2 , α3 , α4 ) =
ΨA−B (α1 , α2 , εi )ΨB (φi , α3 , α4 ),
B
i
where B ∈ H2 (M ), εi runs over a basis for the homology H∗ (M ), and φi is the
dual basis with respect to the intersection pairing. This turns out to be a crucial
ingredient in the proof of associativity of the deformed cup product.
1.7
Quantum cohomology
We show in Chapter 8 how to use the invariants ΦA to define a quantum deformation
of the cup product on the cohomology of a symplectic manifold. To begin with we
shall assume that M is monotone. Moreover, we assume that N ≥ 2 where N is
10
the minimal Chern number defined by h[c1 ], π2 (M )i = N Z. If we multiply ω by
a suitable constant we may assume that λ = 1/N and so h[ω], π2 (M )i = Z.
The basic idea in the definition of quantum cohomology is very easy to understand. Let us write H∗ (M ) and H ∗ (M ) for the integral (co)homology of M modulo
torsion. Thus one can think of H ∗ (M ) as the image of H ∗ (M, Z) in H ∗ (M, R), and
similarly for H ∗ . The advantage of neglecting torsion is that the group H k (M ),
for example, may be identified with the dual Hom(Hk (M ), Z) of Hk (M ). Thus
R a
k-dimensional cohomology class a may be described by specifying all the values γ a
of a on the classes γ ∈ Hk (M ).
We define the quantum multiplication a∗b of classes a ∈ H k (M ) and b ∈ H ` (M )
as follows. Let α = PD(a) and β = PD(b) denote their Poincaré duals. Then a ∗ b
is defined as the formal sum
X
a∗b=
(a ∗ b)A q c1 (A)/N
(1.1)
A
where q is an auxiliary variable, considered to be of degree 2N , and the cohomology
class (a ∗ b)A ∈ H k+`−2c1 (A) (M ) is defined in terms of the Gromov invariant ΦA by
Z
(a ∗ b)A = ΦA (α, β, γ)
γ
for γ ∈ Hk+`−2c1 (A) (M ). Note that the classes α, β, γ satisfy the dimension
condition
2c1 (A) + deg(α) + deg(β) + deg(γ) = 4n
required for the definition of the invariant ΦA . This shows that 0 ≤ c1 (A) ≤ 2n
and hence only finitely many powers of q occur in the formula (1.1). Moreover,
since M is monotone, the classes A which contribute to the coefficient of q d satisfy
ω(A) = c1 (A)/N = d, and hence only finitely many can be represented by Jholomorphic curves. This shows that the sum on the right hand side of (1.1) is
finite. Since only nonnegative exponents of q occur in the sum (1.1) it follows that
a ∗ b is an element of the group
˜ ∗ (M ) = H ∗ (M ) ⊗ Z[q],
QH
where Z[q] is the polynomial ring in the variable q of degree 2N . Extending by
linearity, we therefore get a multiplication
˜ ∗ (M ) ⊗ QH
˜ ∗ (M ) → QH
˜ ∗ (M ).
QH
It turns out that this multiplication is distributive over addition, skew-commutative
and associative. The first two properties are obvious, but the last is much more
subtle and depends on a gluing argument for J-holomorphic curves.
˜ ∗ (M ) vanish for k ≤ 0 and are periodic
The quantum cohomology groups QH
with period 2N for k ≥ 2n. To get full periodicity, one can consider the groups
QH ∗ (M ) = H ∗ (M ) ⊗ Z[q, q −1 ],
where Z[q, q −1 ] is the ring of Laurent polynomials, which consists of all polynomials in the variables q, q −1 with the obvious relation q · q −1 = 1. With this
1.8. NOVIKOV RINGS AND FLOER HOMOLOGY
11
definition, QH k (M ) is nonzero for both positive and negative k, and there is a
natural isomorphism
QH k (M ) → QH k+2N (M ),
given by multiplication with q, for every k ∈ Z.
Note that if A = 0, then all J-holomorphic curves in the class A are constant. It
follows that ΦA (α, β, γ) is just the usual triple intersection α · β · γ. Since ω(A) > 0
for all other A which have J-holomorphic representatives, the constant term in a ∗ b
is just the usual cup product.
As an example, let M be complex projective n-space CP n with its usual Kähler
form. If p is the positive generator of H 2 (CP n ), and if L is the class in H2 represented by the line CP 1 , then the fact that there is a unique line through any two
points is reflected in the identity
Z
(p ∗ pn )L = ΦL ([CP n−1 ], pt, pt) = 1.
pt
Since all the other classes (p ∗ pn )A vanish for reasons of dimension, it follows that
p ∗ pn = q
and hence the quantum cohomology of CP n is given by
˜ ∗ (CP n ) =
QH
Z[p, q]
.
hpn+1 = qi
See Example 8.1.6 for more details. The occurence of the letters p, q is no accident here. In Section 8.3, we describe some recent work of Givental and Kim in
which they interpret the quantum cohomology ring of flag manifolds as the ring of
functions on a Lagrangian variety.
We end the chapter by explaining how the higher Gromov-Witten invariants are
generated by a function S which satisfies the WDVV-equation.
1.8
Novikov rings and Floer homology
If the image Γ of the the Hurewicz homomorphism π2 (M ) → H2 (M ) has rank bigger
than 1 it is possible to modify the definition of quantum cohomology slightly, even in
the monotone case. Roughly speaking, one can count J-holomorphic curves which
represent different homology classes separately and hence extract more information
from the moduli spaces. This leads naturally to tensoring with the Novikov ring
Λω associated to the homomorphism Γ → R induced by ω. This homomorphism is
defined by evaluating the symplectic form ω on a homology class A ∈ Γ ⊂ H2 (M )
and hence measures the energy of the J-holomorphic curves which represent A. The
Novikov ring is a kind of completion of the group ring which in the case where Γ is
1-dimensional and M is monotone agrees with the ring of Laurent power series in
q, q −1 . In the monotone case, one can define the deformed cup product using either
the polynomial ring Z[q] or the Novikov ring as coefficients. However, for more
general symplectic manifolds, for example in the Calabi-Yau case where c1 = 0, it
is necessary to take coefficients in some kind of completed ring, because there may
12
be infinitely many J-holomorphic curves intersecting the cycles α, β, γ, although
there will only be finitely many with energy less than any fixed bound. If we also
want to carry along the information about the homology class A in order to retain as
much information as possible, this leads precisely to the framework of the Novikov
ring. All this is explained in much more detail in Chapter 9.
There is a striking similarity of these structures with Floer homology (cf. [18],
[20]). In the monotone case, the Floer homology groups of the loop space of M are
naturally isomorphic to the the ordinary homology groups of M rolled up with period
2N , which are exactly the quantum cohomology groups in their periodic form, i.e.
tensored with Z[q, q −1 ]. Moreover, in the non-monotone case, the Novikov ring
enters into the construction of Floer homology. This was already pointed out by
Floer in his original paper [20] and was later rediscovered in a joint paper [32] of the
second author with Hofer. There is a natural ring structure on the Floer homology
groups given by the pair-of-pants construction, and one is led to wonder whether
this ring structure agrees with the deformed cup-product. In the last chapter we
shall explain the basic ideas in Floer homology, and give an outline of the proof
why the ring structure should agree with that of quantum cohomology.
Chapter 2
Local Behaviour
In this chapter we establish the basic properties of J-holomorphic curves. This
includes the unique continuation theorem, which asserts that two curves with the
same ∞-jet at a point must coincide, and Proposition 2.3.1, which asserts that a
simple J-holomorphic curve must have injective points. These are essential ingredients in the transversality theory for J-holomorphic curves discussed in Chapter 3.
The other main results are Lemmata 2.2.1 and 2.2.3.
2.1
The generalised Cauchy-Riemann equation
Let (M, J) be an almost complex manifold and (Σ, j) be a Riemann surface. A
smooth map u : Σ → M is called J-holomorphic if the differential du is a complex
linear map with respect to j and J:
J ◦ du = du ◦ j.
Sometimes it is convenient to write this equation as ∂¯J (u) = 0. Here, the 1-form
1
∂¯J (u) = (du + J ◦ du ◦ j) ∈ Ω0,1 (u∗ T M )
2
is the complex
antilinear part of du, and takes values in the complex vector bundle
u∗ T M = (z, v) | z ∈ Σ, v ∈ Tu(z) M . Thus there is an infinite dimensional vector
bundle
E → Map(Σ, M )
whose fibre at u ∈ Map(Σ, M ) is the space Eu = Ω0,1 (u∗ T M ) and ∂¯J is a section
of this bundle. The J-holomorphic curves are the zeros of this section.
As we shall see later, this global form of the equation is useful when one is
discussing properties of the moduli space of all J-holomorphic curves. In order to
study local properties of these curves, it is useful to write the equation in local
coordinates.
By the integrability theorem there exists an open cover {Uα }α of Σ with charts
α : Uα → C such that dα(p)jv = idα(p)v for v ∈ Tp Σ. In particular the transition
maps α◦β −1 are holomorphic. Such local coordinates are called conformal. (Recall
13
14
CHAPTER 2. LOCAL BEHAVIOUR
that a map φ between open subsets of C is conformal, i.e. preserves angles and
orientation, if and only if it is holomorphic.) A smooth map u : Σ → M is Jholomorphic if and only if its local coordinate representations
uα = u ◦ α−1 : α(Uα ) → M
are J-holomorphic with respect to the standard complex structure i on the open
set α(Uα ) ⊂ C.
In conformal coordinates z = s + it on Σ the 1-form ∂¯J (u) is given by
∂uα
1 ∂uα
∂uα
1 ∂uα
+ J(uα )
ds +
− J(uα )
dt.
∂¯J (uα ) =
2 ∂s
∂t
2
∂t
∂s
Hence u is a J-holomorphic curve if and only if in conformal coordinates it satisfies
the nonlinear first order partial differential equation
∂uα
∂uα
+ J(uα )
= 0.
∂s
∂t
(2.1)
In the case of the standard complex structure J = J0 on Cn = R2n equation (2.1)
reduces to the Cauchy-Riemann equations
∂f
∂g
=
,
∂s
∂t
∂f
∂g
=− .
∂t
∂s
for a smooth map u = f +ig : C → Cn . Thus a J0 -holomorphic curve is holomorphic
in the usual sense.
Lemma 2.1.1 Assume Σ is a connected Riemann surface. If two J-holomorphic
curves u, u0 : Σ → M have the same ∞-jet at a point z ∈ Σ then u = u0 .
Proof: Since Σ is connected it suffices to prove this locally. Hence assume that u
and u0 are solutions of (2.1) on some open neighbourhood Ω of 0 ∈ C. Denote by
∆ the standard Laplacian
∆=
∂2
∂
+ 2 = (∂s )2 + (∂t )2 .
2
∂s
∂t
Then it is easy to check that u and u0 are solutions of the second order quasi-linear
equation
∆ u = (∂t J(u))∂s u − (∂s J(u))∂t u.
(2.2)
(Use the fact that ∂t (J 2 ) = (∂t J)J + J(∂t J) = 0.) Now the function v = u0 − u
vanishes to infinite order at 0 ∈ Ω. Because J and its derivatives are bounded it
follows that v satisfies differential inequalities of the form
|∆v(z)| ≤ K(|v| + |∂s v| + |∂t v|),
for all z ∈ Ω. Hence the assertion of the lemma follows from Aronszajn’s unique
continuation theorem which we now quote.
2
2.2. CRITICAL POINTS
15
Theorem 2.1.2 (Aronszajn) Let Ω ⊂ C be a connected open set. Suppose that
2,2
the function v ∈ Wloc
(Ω, Rm ) satisfies the pointwise estimate
∂v
∂v
|∆v(s, t)| ≤ c |v(s, t)| + (s, t) + (s, t)
∂s
∂t
(almost everywhere) and that v vanishes to infinite order at the point z = 0 in the
sense that
Z
|v(z)| = O(rk )
|z|≤r
for every k > 0. Then v ≡ 0.
2,2
Here Wloc
(Ω, Rm ) is the Sobolev space of maps whose second derivative is L2
on each precompact open subset of Ω. Since all the maps which we consider here
are smooth, a reader who is unfamiliar with Sobolev spaces can suppose that g is
C ∞ . This theorem can be viewed as a generalization of the unique continuation
theorem for analytic functions. It is proved by Aronszajn in [1] and by Hartman
and Wintner in [29].
Remark 2.1.3 The assumptions of Lemma 2.1.1 require that the J-holomorphic
curves u and u0 are C ∞ -smooth. This condition is automatically satisfied when the
almost complex structure J is smooth (see Proposition 3.2.2 below). If, however,
J is only of class C ` then the J-holomorphic curves will in general also be only of
class C ` .1 In this case there is an analogue of Lemma 2.1.1 which requires that the
difference v = u0 − u (in local coordinates) vanishes to infinite order at a point z0
(as in the statement of Aronszajn’s theorem). For a proof of this result which does
not rely on Aronszajn’s theorem we refer to [22].
2
2.2
Critical points
A critical point of a J-holomorphic curve u : Σ → M is a point z ∈ Σ such
that du(z) = 0. If we think of the image C = u(Σ) ⊂ M as an unparametrized
J-holomorphic curve, then a critical point on C is a critical value of u. Points on
C which are not critical are called regular or non-singular. In the integrable case
critical points of nonconstant holomorphic curves are well known to be isolated.
The next lemma asserts this for arbitrary almost complex structures.
Lemma 2.2.1 Let u : Σ → M be a nonconstant J-holomorphic curve for some
compact connected Riemann surface Σ. Then the set
X = u−1 ({u(z) | z ∈ Σ, du(z) = 0})
of preimages of critical values is finite.
1 In
fact, a simple elliptic bootstrapping argument based on equation 2.2 shows that if J is of
class C ` with ` ≥ 1 then every J-holomorphic curve u of class W 1,p with p > 2 is necessarily of
class W `+1,q for every q < ∞.
16
Proof: It suffices to prove that critical points are isolated, and so we may work
locally. Thus we may suppose that u is a map of an open neighbourhood Ω of 0 in
C to Cn , which is J-holomorphic for some J : Cn → GL(2n, R), and that
u(0) = 0,
du(0) = 0,
u 6≡ 0,
J(0) = J0 .
Write z = s + it. Since u is non-constant it follows from Lemma 2.1.1 that the
∞-jet of u(z) at z = 0 must be non-zero. Hence there exists an integer ` ≥ 2 such
that u(z) = O(|z|` ) and u(z) 6= O(|z|`+1 ). This implies J(u(z)) = J0 + O(|z|` ).
Now examine the Taylor expansion of
∂u
∂u
+ J(u)
=0
∂s
∂t
up to order ` − 1 to obtain
∂T` (u)
∂T` (u)
+ J0
= 0.
∂s
∂t
Here T` (u) denotes the Taylor expansion of u up to order `. It follows that T` (u) :
C → Cn is a holomorphic function and there exists a nonzero vector a ∈ Cn such
that
∂u
u(z) = az ` + O(|z|`+1 ),
(z) = àz `−1 + O(|z|` ).
∂s
Hence
0 < |z| ≤ ε
=⇒
u(z) 6= 0, du(z) 6= 0
with ε > 0 sufficiently small. Hence preimages of critical values of u are isolated.
2
We now show how to choose nice coordinates near a regular point of a Jholomorphic curve.
Lemma 2.2.2 Let Ω ⊂ C be an open neighbourhood of 0 and u : Ω → M be a local
J-holomorphic curve such that du(0) 6= 0. Then there exists a chart α : U → Cn
defined on a neighbourhood of u(0) such that
α ◦ u(z) = (z, 0, . . . , 0),
dα(u(z))J(u(z)) = J0 dα(u(z))
for z ∈ Ω ∩ u−1 (U ).
Proof: Write z = s + it ∈ Ω and w = (w1 , w2 , . . . , wn ) ∈ Cn where wj = xj + iyj .
Shrink Ω if necessary and choose a complex frame of the bundle u∗ T M such that
Z1 (z), . . . , Zn (z) ∈ Tu(z) M,
Z1 =
∂u
.
∂s
Define φ : Ω × Cn−1 → M by
φ(w1 , . . . , wn ) = expu(w1


n
n
X
X

xj Zj (w1 ) +
yj J(u(w1 ))Zj (w1 ) .
)
j=2
j=2
2.2. CRITICAL POINTS
17
Then φ is a diffeomorphism of a neighbourhood V of zero in Cn onto a neighbourhood U of u(0) in M . It satisfies φ(z1 , 0, . . . , 0) = u(z1 ) and
∂φ
∂φ
+ J(φ)
=0
∂xj
∂yj
at all points z = (z1 , 0, . . . , 0) and j = 1, . . . , n. Hence the inverse α = φ−1 : U → V
is as required.
2
We next start investigating the intersections of two distinct J-holomorphic
curves. The most significant results in this connection occur in dimension 4, and are
discussed in McDuff [44, 48] for example. For now, we prove a useful result which is
valid in all dimensions asserting that intersection points of distinct J-holomorphic
curves u : Σ → M and u0 : Σ0 → M can only accumulate at points which are
critical on both curves C = u(Σ) and C 0 = u0 (Σ0 ). For local J-holomorphic curves
this statement can be reformulated as follows.
Lemma 2.2.3 Let u, v : Ω → M be local non-constant J-holomorphic curves defined on an open neighbourhood Ω of {0} such that
u(0) = v(0),
du(0) 6= 0.
Moreover, assume that there exist sequences zν , ζν ∈ Ω such that
u(zν ) = v(ζν ),
lim zν = lim ζν = 0,
ν→∞
ν→∞
ζν 6= 0 6= zν .
Then there exists a holomorphic function φ : Bε (0) → Ω defined in some neighbourhood of zero such that φ(0) = 0 and
v = u ◦ φ.
Proof: By Lemma 2.2.2 we may assume without loss of generality that M = Cn
and
u(z) = (z, 0),
J(w1 , 0) = i
where w = (w1 , w̃) with w̃ ∈ Cn−1 . Write v(z) = (v1 (z), ṽ(z)).
We show first that the ∞-jet of ṽ at z = 0 must vanish. Otherwise there
would exist an integer ` ≥ 0 such that ṽ(z) = O(|z|` ) and ṽ(z) 6= O(|z|`+1 ). The
assumption of the lemma implies ` ≥ 1 and hence J(v(z)) = J0 + O(|z|` ). As in
the proof of Lemma 2.2.1, consider the Taylor expansion up to order ` − 1 on the
left hand side of the equation ∂s v + J∂t v = 0 to obtain that T` (v) is holomorphic.
Hence
v1 (z) = p(z) + O(|z|`+1 ),
ṽ(z) = ãz ` + O(|z|`+1 )
where p(z) is a polynomial of order ` and a ∈ Cn−1 is nonzero. This implies that
ṽ(z) 6= 0 in some neighbourhood of 0 and hence u(z) 6= v(z) in this neighbourhood,
in contradiction to the assumption of the lemma. Thus we have proved that the
∞-jet of ṽ at z = 0 vanishes.
We prove that ṽ(z) ≡ 0. To see this note that, because J = J0 along the axis
{w̃ = 0},
∂J(w1 , 0)
∂J(w1 , 0)
=
= 0,
∂x1
∂y1
18
for all w1 . Hence
∂J(w) ∂J(w) +
∂x1 ∂y1 ≤ c |w̃| .
As in Lemma 2.1.1, it follows easily that
|∆ṽ| ≤ c(|ṽ| + |∂s ṽ| + |∂t ṽ|).
Hence it follows from Aronszajn’s theorem that ṽ ≡ 0. The required function φ is
now given by φ(z) = v1 (z).
2
2.3
Somewhere injective curves
A curve u : Σ → M is said to be multiply-covered if it is the composite of a
holomorphic branched covering map (Σ, j) → (Σ0 , j 0 ) of degree greater than 1 with
a J-holomorphic map Σ0 → M . The curve u is called simple if it is not multiply
covered. We shall see in the next chapter that the simple J-holomorphic curves
in a given homology class form a smooth finite dimensional manifold for generic
J. In other words, the multiply covered curves are the exceptional case and they
may be singular points in the moduli space of J-holomorphic curves. The proof
of this result is based on the observation that every simple J-holomorphic curve is
somewhere injective in the sense that
du(z) 6= 0,
u−1 (u(z)) = {z}
for some z ∈ Σ. A point z ∈ Σ with this property is called an injective point for
u.
Proposition 2.3.1 Every simple J-holomorphic curve u is somewhere injective.
Moreover the set of injective points is open and dense in Σ.
Proof: Let X be the finite set of critical points of u and
X 0 = u(X)
be the corresponding set of critical values. Further, let Q be the set of points of
Y = u(Σ) − X 0
where distinct branches of u(Σ) meet. By Lemma 2.2.1, Q is a discrete subset of Y
(i.e. it has no accumulation points in Y ). Thus the set S = Y − Q is an embedded
submanifold in M . Let ι : S → M denote this embedding. Since only finitely many
branches of Y can meet at each point of Q, each such point gives rise to a finite
number of ends of S each diffeomorphic to a deleted disc D − pt. Therefore, we
may add a point to each of these ends and extend ι smoothly over the resulting
surface S 0 to ι0 . Because ι0 is an immersion, there is a unique complex structure on
S 0 with respect to which ι0 is J-holomorphic.
The manifold S 0 still has ends corresponding to the points in X 0 . Each such
end corresponds to a distinct branch of Im u through a point in X 0 . Thus it is the
conformal image of a deleted disc, and so must have the conformal structure of
2.3. SOMEWHERE INJECTIVE CURVES
19
the deleted disc. Therefore we may form a closed Riemann surface Σ0 by adding a
point to each end of S 0 . Further, because u extends over the whole of Σ, the map
ι0 must extend to a J-holomorphic map u0 : Σ0 → M . This map u0 is somewhere
injective and u factors as u0 ◦ ψ where ψ is a holomorphic map Σ → Σ0 . Thus ψ is
a branched cover and has degree 1 iff u is somewhere injective.
In the case Σ = Σ0 = CP 1 the above argument can be rephrased in more explicit
terms as follows. By Lemma 2.2.2 the set
X ⊂ CP 1
of preimages of critical values of u is finite. Denote by
Γ0 ⊂ (CP 1 − X) × (CP 1 − X)
the set of all pairs (z, ζ) such that there exist sequences zν → z and ζν → ζ with
u(zν ) = u(ζν ) and (zν , ζν ) 6= (z, ζ). In other words, Γ0 is the set of accumulation
points of multiple points of u. Isolated self-intersection points are excluded. It
follows from Lemma 2.2.3 that Γ0 is an equivalence relation on CP 1 − X. The
projection
π : Γ0 → CP 1 − X
onto the first component is a covering and π −1 (z) is a finite set for every z. This
is because u(ζ) = u(z) for every ζ ∈ π −1 (z) and u(z) is a regular value of u in the
sense that du(ζ) 6= 0 for every ζ ∈ u−1 (u(z)). Hence it follows from Lemma 2.2.3
that u−1 (u(z)) is a finite set. Since CP 1 − X is connected the number
k = # ζ ∈ CP 1 − X | (z, ζ) ∈ Γ0
is independent of z. Moreover, it follows from Lemma 2.2.3 that each local inverse
of π is holomorphic.
If k = 1 then it follows from the definition of Γ0 that the set of noninjective
points of u is countable and can accumulate only at the critical set X. In particular
u is somewhere injective in this case. Hence assume k ≥ 2 and extend Γ0 to its
closure
Γ = cl(Γ0 ) ⊂ CP 1 × CP 1 .
This set is an equivalence relation on CP 1 and we denote
z∼ζ
⇐⇒
(z, ζ) ∈ Γ.
The set X is invariant under this equivalence relation. (If z ∈ X and z ∼ ζ then
ζ ∈ X.) Hence each point z ∈ CP 1 carries a natural multiplicity m(z) ≥ 1 defined
as follows. If z ∈ CP 1 − X define m(z) = 1. If z ∈ X and w ∈ CP 1 − X is
sufficiently close to z then all k points in the equivalence class of w are close to
X. Define m(z) as the number of points equivalent to w which are close to z. By
continuity this number is independent of the choice of w. With this definition we
have
X
m(ζ) = k
ζ∼z
1
for every z ∈ CP .
20
We prove that the holomorphic equivalence relation Γ gives rise to a meromorphic map ψ : CP 1 → CP 1 of degree k such that
⇐⇒
ψ(z1 ) = ψ(z2 )
z1 ∼ z2 .
To see this define p : C × C → C by
Q
p(z1 , z2 ) =
ζ∼z2 (z1
− ζ)m(ζ)
z1 − z2
Then p is a polynomial of degree k − 1 in z1 and an entire function in z2 . Since the
entire function z2 7→ p(z1 , z2 ) has precisely k − 1 zeros it must be a polynomial of
degree k − 1. Since these zeros agree with those of z2 7→ p(z2 , z1 ) both polynomials
agree up to a constant factor. Examining the case z1 = z2 we find
p(z1 , z2 ) = p(z2 , z1 ).
Now p(z1 , z2 ) = 0 iff z1 ∼ z2 and z1 6= z2 . This implies
p(z1 , z2 ) =
ψ(z1 ) − ψ(z2 )
z1 − z2
for some polynomial ψ : C → C of degree k. To see this choose ψ(z) = zp(z, 0)
and examine the zeros on both sides of the above equation. This function ψ is as
required. It is a holomorphic map of degree k such that ψ(z1 ) = ψ(z2 ) implies
u(z1 ) = u(z2 ). Hence there exists a unique map u0 : CP 1 → M such that
u = u0 ◦ ψ.
The map u0 is obviously continuous on CP 1 and J-holomorphic on the complement
of X. It follows again from the removable singularity theorem for J-holomorphic
curves that u0 is smooth.
2
Proposition 2.3.2 Let uj : CP 1 → M be simple J-holomorphic curves for j =
1, . . . , N such that ui 6= uj ◦ φ for i 6= j and for any fractional linear transformation
φ. Then there exist points z1 , . . . , zN ∈ CP 1 such that
duj (zj ) 6= 0,
u−1
j (uj (zj )) = {zj }
for all j and
ui (zi ) ∈
/ uj (CP 1 )
for i 6= j. Moreover, the set of all N -tuples (z1 , . . . , zN ) ∈ (CP 1 )N which satisfy
these conditions is open and dense in (CP 1 )N and its complement has codimension
at least 2.
Proof: Since the curves uj are simple it follows from Proposition 2.3.1 that there
exist points zj ∈ CP 1 which are injective for uj . In fact the proof of Proposition 2.3.1 shows that there are at most countably many points z1 which are not
injective for u1 in the above sense and these can only accumulate only at the finitely
many singular points of u1 .
2.3. SOMEWHERE INJECTIVE CURVES
21
Now we shall prove that for every pair i 6= j the set of N -tuples (z1 , . . . , zN )
with ui (zi ) 6= uj (CP 1 ) and uj (zj ) ∈
/ ui (CP 1 ) is open and dense in (CP 1 )N . To
1
see this denote by Yj ⊂ CP the set of points z ∈ CP 1 such that either uj (z) is
a critical value of uj or uj (z) = uj (ζ) for some ζ 6= z. In other words, Yj is the
complement of the set of injective points for uj and hence, by Proposition 2.3.1, it
is countable with only finitely many possible accumulation points. Now denote by
Γij ⊂ (CP 1 − Yi ) × (CP 1 − Yj )
the set of all pairs (zi , zj ) such that there exist sequences ziν → zi and zjν → zj
with ui (ziν ) = uj (zjν ) and (ziν , zjν ) 6= (zi , zj ). In other words, Γij is the set
of accumulation points of intersections of ui and uj . In this case Γij is not an
equivalence relation but it follows as before from Lemma 2.2.3 that the projections
πi : Γij → CP 1 − Yi ,
πj : Γij → CP 1 − Yj
are coverings with locally holomorphic inverses. Moreover, as before the number of
points in the fiber πi−1 (z) is independent of z and, by definition of Yi , it is either 0
or 1.
In the case π −1 (z) = ∅ it follows from the definition of Γij that the set of
points z ∈ CP 1 with ui (z) ∈ uj (CP 1 ) is countable with finitely many possible
accumulation points and hence its complement is open and dense in CP 1 . Since
this holds for every pair (i, j) with i 6= j this proves the proposition. Hence assume
that the set πi−1 (z) consists of precisely one point for each z ∈ CP 1 − Yi . Then
there is a unique map φ : CP 1 − Yi → CP 1 − Yj such that
(z, φ(z)) ∈ Γij
for all z ∈ CP 1 − Yi and hence
ui = uj ◦ φ.
Now the map φ is holomorphic and, as in the proof of Proposition 2.3.1, it extends
to a biholomorphic transformation of CP 1 . By assumption such a map does not
exist and this proves the proposition.
2
It is worth pointing out that all the results of the last two sections remain valid
for almost complex structures of class C ` with ` ≥ 2. In particular, this holds for
the unique continuation theorem, in view of Remark 2.1.3. In the next chapter we
shall use the C ` versions of these results to prove the transversality theorems for
J-holomorphic curves.
Another important group of results concerns the positivity of intersections of
J-holomorphic curves with J-holomorphic submanifolds of codimension 2. This
is especially important in the study of symplectic 4-manifolds. These results are
not relevant for the theorems proved in this book although they might play an
important role in computing quantum cohomology groups or Gromov-invariants in
specific cases. We refer the interested reader to [50] and [53].
22
Chapter 3
Moduli Spaces and
Transversality
Let (M, J) be a 2n-dimensional almost complex manifold without boundary and
let Σ be a closed Riemann surface of genus g with complex structure j. In this
chapter we examine the space
M(A, J)
of all simple J-holomorphic curves which represent a given homology class A ∈
H2 (M ). For simplicity, we will assume that the domain is a fixed Riemann surface
(Σ, j). We begin by outlining a proof of the fact that the space M(A, J) is a
finite dimensional manifold for a “generic” set Jreg (A) of almost complex structures
J. For simplicity we shall assume throughout that our symplectic manifold M is
compact. However, it is easy to see that all the results of this chapter generalize to
the noncompact case.
3.1
The main theorems
Denote by X = Map(Σ, M ; A) the space of all smooth maps u : Σ → M which are
somewhere injective and represent the homology class A ∈ H2 (M ). This space can
be thought of as a kind of infinite dimensional manifold whose tangent space at
u ∈ X is the space
Tu X = C ∞ (u∗ T M )
of all smooth vector fields ξ(z) ∈ Tu(z) M along u. Consider the infinite dimensional
vector bundle E → X whose fiber at u is the space
Eu = Ω0,1 (u∗ T M )
of smooth J-anti-linear 1-forms with values in u∗ T M . Recall that the complex
anti-linear part of du defines a section
∂¯J : X → E
of this vector bundle and the moduli space M(A, J) = ∂¯J−1 (0) is the intersection
with the zero section. In order for M(A, J) to be a manifold it is required that ∂¯J
23
24
CHAPTER 3. MODULI SPACES AND TRANSVERSALITY
be transversal to the zero section. This means that the image of the linearization
d∂¯J (u) : Tu X → T(u,0) E of ∂¯J is complementary to the tangent space Tu X of the
zero section. Let us write D∂¯J (u) for the composite of d∂¯J (u) with the projection
πu : T(u,0) E = Tu X ⊕ Eu → Eu . Then we require that the linearized operator
Du = D∂¯J (u) : C ∞ (u∗ T M ) → Ω0,1 (u∗ T M )
is surjective for every u ∈ M(A, J). Formally the operator Du can be determined
by differentiating the local co-ordinate expressions for ∂¯J (u) in the direction of a
vector field ξ along u. These expressions define a global complex anti-linear 1-form
on u∗ T M whenever u is a J-holomorphic curve.
It turns out that Du is an elliptic first order partial differential operators and so
is Fredholm. This means that Du has a closed range and finite dimensional kernel
and cokernel. The Fredholm index of such an operator is defined as the dimension of
the kernel minus the dimension of the co-kernel. This index is stable under compact
(lower order) perturbations. Now C ∞ (u∗ T M ) and Ω0,1 (u∗ T M ) are complex vector
spaces but, due to the nonintegrability of J, the operator Du does not respect the
complex structures. However, the complex anti-linear part of the operator Du is
of lower order and removing it does not change the Fredholm index. The complex
linear part is a Cauchy-Riemann operator and determines a holomorphic structure
on u∗ T M . The index of such operators is given by the Riemann-Roch theorem and
we obtain
index Du = n(2 − 2g) + 2c1 (u∗ T M ).
If the operator Du is onto for every u ∈ M(A, J), it follows from the infinite
dimensional implicit function theorem that the space M(A, J) is indeed a finite
dimensional manifold whose tangent space at u is the kernel of Du . We will suppose
that the almost complex structure J varies in some space J of almost complex
structures on M which is sufficiently large for the constructions given below to
work. For example, J could be any subset of the space of all smooth structures
which is open in the C ∞ -topology or, if M has a symplectic form ω, it could be the
space of all ω-compatible structures. In all cases J carries the usual C ∞ -topology.
Definition 3.1.1 A point (u, J) is called regular if Du is onto. Given J as above
and A ∈ H2 (M, Z) we denote by Jreg (A) the set of all J ∈ J such that Du is onto
for every u ∈ M(A, J). We also denote by
\
Jreg =
Jreg (A)
A
the set of all J ∈ J such that Du is onto for every simple J-holomorphic curve
u : CP 1 → M . We shall use the notation Jreg = Jreg (M, ω) when J (M, ω) is the
space of ω-compatible almost complex structures. (This is defined in Section 3.2
below).
The first part of the theorem below follows immediately from this definition
together with the implicit function theorem proved in Section 3.3. The second part
uses a version of Sard’s theorem which will be discussed later on.
Theorem 3.1.2 (i) If J ∈ Jreg (A) then the space M(A, J) is a smooth manifold
of dimension n(2 − 2g) + 2c1 (A). It carries a natural orientation.
3.2. ELLIPTIC REGULARITY
25
(ii) The set Jreg (A) has second category in J . This means that it contains the
intersection of countably many open and dense subsets of J .
The next task is to discuss the dependence of the manifolds M(A, J) on the
choice of J ∈ Jreg (A). A smooth homotopy of almost complex structures is a
smooth map [0, 1] → J : λ → Jλ . For any such homotopy define
M(A, {Jλ }λ ) = {(λ, u) | u ∈ M(A, Jλ )} .
Given J0 , J1 ∈ Jreg denote by J (J0 , J1 ) the space of all smooth homotopies of
almost complex structures connecting J0 to J1 . In general, even if J is pathconnected, there does not exist a homotopy such that Jλ ∈ Jreg for every λ. In
other words the space M(A, Jλ ) may fail to be a manifold for some values of λ.
However, there is always a smooth homotopy such that the space M(A, {Jλ }λ ) is
a manifold. Such homotopies are called regular. Intuitively speaking, one can
think of the space Jreg of regular almost complex structures as the complement of
a subvariety S of codimension 1 in the space J . A smooth homotopy λ 7→ Jλ is
regular if it is transversal to S. For regular homotopies the space M(A, {Jλ }λ ) is
a manifold with boundary
∂M(A, {Jλ }λ ) = M(A, J1 ) − M(A, J0 ).
The minus sign indicates the reversed orientation.
Theorem 3.1.3 Assume that J is path-connected and consider J0 , J1 ∈ Jreg (A).
Then there exists a dense set of homotopies
Jreg (A, J0 , J1 ) ⊂ J (J0 , J1 )
such that, for every {Jλ }λ ∈ Jreg (A, J0 , J1 ), the space M(A, {Jλ }λ ) is a smooth
manifold of dimension n(2 − 2g) + 2c1 (A) + 1. This manifold carries a natural
orientation.
Thus the two moduli spaces M(A, J1 ) and M(A, J0 ) are oriented cobordant.
Note, however, that until we establish some version of compactness this does not
mean very much. The problem of compactness will be addressed in Chapter 4.
That chapter is more or less independent of the present one. However, in order
to understand our notation and to have some perspective on Sobolev spaces, we
recommend that the reader continue at least until the end of the next section.
3.2
Elliptic regularity
In preparation for the proofs of Theorems 3.1.2 and 3.1.3 we shall first introduce
the appropriate Sobolev space framework and discuss (without proof) the relevant
elliptic regularity theorems. We shall then prove the first part of Theorem 3.1.2
under the assumption that the operator Du is surjective for every u ∈ M(A, J).
In the final and main part of this chapter we shall then address the question of
transversality.
We shall work with the space J = J (M, ω) of all almost complex structures on
M which are compatible with some symplectic form ω on M . This means that
26
(i) ω(v, Jv) > 0 for all non-zero tangent vectors v ∈ T M , and
(ii) ω(Jv, Jw) = ω(v, w) for all v, w ∈ T M .
An equivalent condition is that the formula
hv, wi = ω(v, Jw)
(3.1)
defines a Riemannian metric on M . In fact there is a unique Hermitian form on
T M whose real part is given by (3.1) and whose imaginary part is the symplectic
form ω. Conversely, any such Hermitian structure on M with imaginary part ω
determines a compatible almost complex structure. It is well known that the space
J (M, ω) is non-empty and contractible. (See, for example, [52].)
Remark 3.2.1 We could equally well work with the larger space Jτ (M, ω) of ωtame J, consisting of all almost complex structures which satisfy condition (i) above.
The proof for this case is slightly simpler than in the ω-compatible case. When it
comes to the question of compactness, we shall work with tame J since now the
result for tame J implies that for compatible J. In later chapters it will sometimes
be convenient to blur the distinction between them and write J (M, ω) to denote
either space.
Later on it will be important to consider almost complex structures of class
C ` , rather than smooth ones, in order to obtain a parameter space with a Banach
manifold structure. Hence we shall consider the space
J ` = J ` (M, ω)
of all almost complex structures of class C ` which are compatible with ω. We will
always assume ` ≥ 1 so that in order for the terms in the equation (2.1) to be
well-defined functions. Moreover, we shall denote by
X k,p
the space of maps Σ → M whose k-th derivatives are of class Lp and which represent
the class A ∈ H2 (M, Z). As explained in more detail in Appendix B, this is the
completion of X with respect to the Sobolev W k,p -norm given by the sum of the
Lp -norms of all derivatives of u up to order k. These norms can be defined in
terms of covariant derivatives using Riemannian metrics on M and Σ. Since all our
manifolds are compact this norm does not depend on the choice of these metrics.
In order for the space X k,p to be well defined we must assume that
kp > 2.
There are various reasons for this. Firstly, the very definition of X k,p , in terms of local coordinate representations of class W k,p , requires this assumption. The point is
this: the W k,p -norm is well-defined for maps between open sets in Euclidean space,
but for a general manifold one needs a space which is invariant under composition
with coordinate charts. Now the composition of a W k,p -map u : R2 → R2n with
a C k -diffeomorphism in the source and a C k -map in the target is again of class
W k,p precisely when kp > 2. Secondly, the Sobolev embedding theorem asserts
3.3. IMPLICIT FUNCTION THEOREM
27
that under this condition the space X k,p embeds into the space of continuous maps
from Σ → M and, by Rellich’s theorem, this embedding is compact. Thirdly, the
condition kp > 2 is required to obtain that the product of two W k,p -functions is
again of this class. In other words, the condition kp > 2 is needed to deal with the
nonlinearities.
The first key observation is that every J-holomorphic curve of class W 1,p with
p > 2 in a smooth manifold is necessarily smooth. More precisely, we have the
following regularity theorem which can be proved by elliptic bootstrapping methods.
The details are carried out in Appendix B.
Proposition 3.2.2 (Elliptic regularity) Assume J ∈ J ` is an almost complex
structure of class C ` with ` ≥ 1. If u : Σ → M is a J-holomorphic curve of class
W 1,p with p > 2 then u is of class W `+1,p . Moreover, there is a constant c = c(J, `)
such that
kukW `+1,p ≤ ckukW 1,p .
In particular, u is of class C ` , and if J is smooth (C ∞ ) then so is u.
This shows that for J ∈ J ` the moduli space of J-holomorphic curves of class
W
is independent of the choice of k so long as k ≤ ` + 1. In fact this condition
k ≤ ` + 1 is needed for the operator Du to be well defined on the appropriate
Sobolev spaces.
k,p
Remark 3.2.3 An important consequence of elliptic regularity is the fact that the
kernel and cokernel of an elliptic operator do not depend on the precise choice of
the space on which the operator is defined. In our context assume that J is of
class C ` and u is a J-holomorphic curve and so is of class W `+1,q for any q. Now
consider the operator
Du : W k,p (u∗ T M ) → W k−1,p (Λ0,1 Σ ⊗J u∗ T M )
with k ≤ ` + 1.1 Then every element in the kernel of Du is necessarily of class
W `+1,q for any q and so the kernel of Du does not depend on the choice of k and
p as long as k ≤ ` + 1. Similar remarks apply to the cokernel. In particular, the
operator Du is onto for some choice of k and p if and only if it is onto for all such
choices.
2
3.3
Implicit function theorem
In this section we shall formulate an appropriate version of the implicit function
theorem. Roughly speaking, this theorem asserts that if u is an approximate Jholomorphic curve, in the sense that ∂¯J (u) is sufficiently small in the Lp -norm,
and if the operator Du is surjective with a uniformly bounded right inverse then
1 Given a vector bundle E → M , we write C ∞ (E) for the space of C ∞ -smooth sections and
W k,p (E) for its completion with respect to the W 1,p -norm, i.e. the k-th derivatives of W k,p sections are of class Lp . Further Λ0,1 Σ denotes the bundle of 1-forms on Σ of type 0, 1. Hence
C ∞ (Λ0,1 Σ ⊗J u∗ T M ) = Ω0,1 (u∗ T M ).
28
there exists an actual J-holomorphic curve near u. This theorem will also play an
important role in the gluing construction in Appendix A and hence we shall explain
it carefully. This section may be skipped at a first reading. The next section may
be read independently. We shall assume throughout this section that J is a fixed
smooth almost complex structure.
In view of Proposition 3.2.2 it suffices to work with the Sobolev space X 1,p
of W 1,p -maps u : Σ → M for some fixed number p > 2. Consider the infinite
dimensional Banach space bundle E p → X 1,p whose fiber at u ∈ X 1,p is the space
Eup = Lp (Λ0,1 T ∗ Σ ⊗J u∗ T M )
of complex anti-linear 1-forms on Σ of class Lp which take values in the pullback
tangent bundle u∗ T M . The nonlinear Cauchy-Riemann equations determine a
section
∂¯J : X 1,p → E p
of this bundle whose differential at a J-holomorphic curve u : Σ → M is the
operator
Du : W 1,p (u∗ T M ) → Lp (Λ0,1 T ∗ Σ ⊗J u∗ T M ).
This operator is uniquely determined for each J-holomorphic curve u, however its
definition for general u depends on a choice of connection on T M . If we take the
Levi-Civita connection ∇ of the metric (3.1) which is induced by J then an explicit
formula for Du is
Du ξ = 12 (∇ξ + J(u)∇ξ ◦ j) + 12 (∇ξ J)(u)∂J (u) ◦ j.
(3.2)
An alternative formula in terms of a Hermitian connection and the Nijenhuis tensor
is given below. The two formulae only agree for J-holomorphic curves u and will
be different for other maps.
Remark 3.3.1 (Formula for Du ) Here is another formula for Du for a curve u
which is J-holomorphic. We will write it in terms of a Hermitian connection ∇ on
M , so that parallel translation commutes with J. When J is not integrable, we
cannot assume that ∇ is torsion free. However, as is shown in [34], we may assume
that it equals 41 NJ , where NJ is the Nijenhuis torsion tensor which measures the
non-integrability of J. In this notation, the operator
Du : W k,p (u∗ T M ) → W k−1,p (Λ0,1 T ∗ Σ ⊗J u∗ T M )
is given by the following formula
Du ξ =
1
2
(∇ξ + J(u)∇ξ ◦ j) + 18 NJ (∂J (u), ξ).
A geometric derivation of this formula may be found in McDuff [42], Proposition 4.1,
for example.
2
The ellipticity of the operator Du can be expressed in terms of the estimate
kξkW 1,p ≤ c0 (kDu ξkLp + kξkLp )
(3.3)
This estimate follows from the Calderon-Zygmund inequality (the Lp -estimate for
Laplace’s operator) which is proved in Appendix B.
29
Remark 3.3.2 The constant c0 in (3.3) depends on the choice of a metric on the
Riemann surface Σ. In Appendix A we shall see that it is sometimes important to
choose a nonstandard metric with part of the volume concentrated in an arbitrarily
small ball. But the volume of Σ with respect to all these metrics will be uniformly
bounded. Moreover, these metrics are chosen such that the Lp norm of the 1-form
du ∈ Ω1 (u∗ T M ) satisfies a uniform bound.
2
The estimate (3.3) implies that the operator Du has a closed range and a finite
dimensional kernel. Now consider the formal adjoint operator
Du∗ : W 1,q (Λ0,1 T ∗ Σ ⊗J u∗ T M ) → Lq (u∗ T M ).
where 1/p + 1/q = 1. This operator is defined by the identity
hη, Du ξi = hDu∗ η, ξi
for ξ ∈ C ∞ (u∗ T M ) and η ∈ Ω0,1 (u∗ T M ). Integration by parts shows that Du∗ is
again a first order elliptic operator (with coefficients in the same class as u) and
hence satisfies an estimate (3.3) with p replaced by q. Hence the operator Du∗ also
has a finite dimensional kernel and a closed range. Moreover, it follows from elliptic
regularity that every η ∈ Lq (Λ0,1 T ∗ Σ ⊗J u∗ T M ) which annihilates the range of Du
(i.e. hη, Du ξi = 0 for all ξ ∈ W 1,p (u∗ T M )) is in fact of class W 1,q and in the kernel
of Du∗ . (See Exercise B.3.5 in Appendix B.) This shows that the operator Du has
a finite dimensional cokernel.
Remark 3.3.3 (Du∗ in local coordinates) Choose conformal coordinates z =
s + it on Σ and assume that the Riemannian metric on Σ is in these coordinates
given by θ−2 (ds2 + dt2 ) for a smooth function θ = θ(z). For example the FubiniStudy metric on Σ = CP 1 is in the standard coordinates given by θ(z) = 1 + |z|2 .
In such coordinates a complex anti-linear 1-form η ∈ Ω0,1 (u∗ T M ) can be written
in the form
η = ζ ds − Jζ dt
where ζ(z) ∈ Tu(z) M . Recall from (3.2) that the operator Du is given by the
formula
Du ξ = ζ ds − Jζ dt,
ζ = 21 ∇s ξ + J∇t ξ + 12 (∇ξ J)(∂t u + J∂s u)
where ∇ denotes the Levi-Civita connection of the J-induced metric. A simple
calculation shows that the formal adjoint operator is given by
Du∗ η = θ2 −∇s ζ + J∇t ζ + 12 (∇ζ J)(∂t u + J∂s u) + 12 (∇∂t u−J∂s u J)ζ .
2
We shall now turn to the question of surjectivity of the operator Du . In view of
the preceding discussion this is equivalent to the injectivity of the adjoint operator
Du∗ . For the implicit function theorem it is important to have a quantitative expression of surjectivity. Roughly speaking, this means that the norm of a suitable
right inverse does not get too large. One possibility for constructing such a right
inverse is to take the operator
Qu = Du∗ (Du Du∗ )−1 : Lp (Λ0,1 T ∗ Σ ⊗J u∗ T M ) → W 1,p (u∗ T M )
30
One can prove that an inequality of the form kηkW 1,q ≤ c kDu∗ ηkLq implies a similar
inequality for the operator Du restricted to the image of Du∗ , a complement of the
kernel of Du . (See for example Lemma 4.5 in [16].) An alternative technique
for constructing a right inverse is to reduce the domain W 1,p (u∗ T M ) of Du , by
imposing pointwise conditions on ξ, so that the resulting opoerator is bijective, and
then taking Qu to be the inverse of this restricted operator.
The next theorem is a refined version of the implicit function theorem discussed
in the beginning of this section. We actually prove that if u is an approximate
J-holomorphic curve with sufficiently surjective operator Du , then there are Jholomorphic curves near u and they can be modelled on a neighbourhood of zero
in the kernel of Du . More explicitly, for every sufficiently small ξ ∈ ker Du we can
find a unique J-holomorphic curve of the form vξ = expu (ξ + Qu η).
Theorem 3.3.4 Let p > 2 and 1/p + 1/q = 1. Then for every constant c0 > 0
there exist constants δ > 0 and c > 0 such that the following holds. Let u : Σ → M
be a W 1,p -map and Qu : Lp (Λ0,1 T ∗ Σ ⊗J u∗ T M ) → W 1,p (u∗ T M ) be a right inverse
of Du such that
kQu k ≤ c0 , kdukLp ≤ c0 , ∂¯J (u)Lp ≤ δ
with respect to a metric on Σ such that Vol(Σ) ≤ c0 . Then for every ξ ∈ ker Du
with kξkLp ≤ δ there exists a section ξˆ = Qu η ∈ W1,p (u∗ T M ) such that
∂¯J (expu (ξ + Qu η)) = 0,
kQu ηkW1,p ≤ c ∂¯J (expu (ξ))Lp .
Proof: The proof is an application of the implicit function theorem for the map
F : W 1,p (u∗ T M ) → Lp (Λ0,1 T ∗ Σ ⊗J u∗ T M )
defined by
F(ξ) = Φξ (∂¯J (expu (ξ)))
(3.4)
where
p
Φξ : Eexp
u (ξ)
→ Eup
denotes parallel transport along the geodesic τ 7→ expu (τ ξ). The map F is smooth
and its derivatives are controlled by the Lp -norm of du. The differential at zero is
given by
dF(0) = Du
and our assumptions guarantee that this operator is onto and has a right inverse
Qu : Lp (Λ0,1 T ∗ Σ ⊗J u∗ T M ) → W 1,p (u∗ T M ) such that
Du Qu = 1l,
kQu k ≤ c0 .
Moreover the function F satisfies a quadratic estimate of the form
F(ξ + ξˆ ) − F(ξ) − dF(ξ)ξb p ≤ c1 ξˆ ∞ ξˆ 1,p .
L
L
(3.5)
W
Putting these estimates together we can find the required solution ξˆ = Qu η of
F(ξ + Qu η) = 0
31
by the contraction mapping principle for the contraction η 7→ η − F(ξ + Qu η). The
solution is the limit of the sequence ην generated by the standard iteration
ην+1 = ην − F(ξ + Qu ην )
with η0 = 0. With ξν = ξ + Qu ην this can also be interpreted as the Newton type
iteration ξν+1 = ξν − Qu F(ξν ). Details of the convergence proof are standard and
are left to the reader.
2
We next show that the J-holomorphic curve which we constructed in the previous theorem from a given ξ is actually unique within a C 0 -neighbourhood of
u. In the following proposition the map u again plays the role of an approximate
J-holomorphic curve, and one should think of v0 = expu (ξ + ξb) as the solution
constructed from ξ.
Proposition 3.3.5 Let p > 2 and 1/p + 1/q = 1. Then for every constant c0 > 0
there exists a constant δ > 0 such that the following holds. Let u : Σ → M be a
W 1,p -map and Qu : Lp (Λ0,1 T ∗ Σ ⊗J u∗ T M ) → W 1,p (u∗ T M ) be a right inverse of
Du such that
kQu k ≤ c0 , kdukLp ≤ c0
with respect to a metric on Σ such that Vol(Σ) ≤ c0 . If v0 = expu (ξ0 ) and v1 =
expu (ξ1 ) are J-holomorphic curves such that ξ0 , ξ1 ∈ W 1,p (u∗ T M ) satisfy
kξ0 kW 1,p ≤ δ,
kξ1 kW 1,p ≤ c0 ,
and
kξ1 − ξ0 kL∞ ≤ δ,
ξ1 − ξ0 ∈ im Qu ,
then v0 = v1 .
Proof: Choose η = Du ξˆ ∈ Lp (Λ0,1 T ∗ Σ ⊗J u∗ T M ) so that
ξˆ = ξ1 − ξ0 = Qu η
ˆ Let ξ 7→ F(ξ) be the map defined in the proof of Theand note that Du η = ξ.
orem 3.3.4. Then F(ξ0 ) = F(ξ1 ) = 0. We use the quadratic estimate (3.5) to
obtain
b
ξ 1,p = kQu ηkW 1,p
W
≤ c0 kηkLp
= c0 Du ξˆ p
L
= c0 dF(0)ξˆ Lp
= c0 F(ξ1 ) − F(ξ0 ) − dF(0)ξˆ Lp
≤ c0 F(ξ0 + ξˆ ) − F(ξ0 ) − dF(ξ0 )ξˆ Lp
+ c0 (dF(ξ0 ) − dF(0))ξˆ Lp
ˆ ˆ
+ c2 δ ξˆ ≤ c0 c1 ξ ξ ∞
1,p
W
W 1,p
L
ˆ
≤ c3 δ ξ 1,p .
W
32
If c3 δ < 1 then ξ1 − ξ0 = ξˆ = 0 and this proves the proposition.
2
Proof of Theorem 3.1.2 (i): If u ∈ M(A, J) and and J ∈ Jreg (A) then, by
assumption, the Fredholm operator Du is surjective. Hence u satisfies the requirements of Theorem 3.3.4 and hence there exists a unique map
f : ker Du → im Du∗
such that
∂¯J (expu (ξ + f (ξ))) = 0
for every sufficiently small section ξ ∈ ker Du . Moreover, the map f is smooth and,
by Proposition 3.3.5, every J-holomorphic curve v, which is bounded in W 1,p and
sufficiently close to u in the L∞ -norm, is of the form
v = expu (ξ + f (ξ))
with ξ ∈ ker Du . To see this write v = expu (ζ) where ζ ∈ W 1,p (u∗ T M ) with
kζkW 1,p ≤ c0 . Now write ζ = ξ + Du∗ η with Du ξ = 0. Then, by Lemma 4.3.1, the
W 1,p -norm of ξ is controlled by its Lp -norm and is therefore small. Hence we can
apply Proposition 3.3.5 to obtain Du∗ η = f (ξ).
Thus we have proved that the map
ker Du → M(A, J) : ξ 7→ expu (ξ + f (ξ))
defines a local coordinate chart of the Moduli space M(A, J). We leave it to the
reader to check that the transition functions obtained from two nearby elements u
and v are smooth. This proves that the moduli space M(A, J) is a smooth manifold
whose dimension agrees with the dimension of the kernel of Du and therefore with
the index of Du which is n(2 − 2g) + 2c1 (A).
Orientations
To understand why the moduli space has a canonical orientation, observe first that
the tangent space Tu M(A, J) is just the kernel of the operator Du . Now, according
to Remark 3.3.1, the operator Du is the sum of two parts: ∇ξ + J(u)∇ξ ◦ j and
1
4 NJ (∂J (u), ξ). The first of these has order 1 and commutes with J, while the
second has order 0 and anti-commutes with J. Hence the kernel of Du will in
general not be invariant under J and so J might not determine an almost complex
structure on Tu M(A, J). However, by multiplying the second part of Du by a
constant which tends to 0, one can homotop Du through Fredholm operators to a
Fredholm operator which does commute with J. The resulting operator
C ∞ (u∗ T M ) → Ω0,1 (u∗ T M ) : ξ 7→ ∂¯u,∇ ξ = ∇ξ + J(u)∇ξ ◦ j
¯
is precisely the Dolbeault ∂-operator.
It determines a holomorphic structure on the
∗
complex vector bundle u T M and its index
∗
index ∂¯u,∇ = dim H∂0¯ (Σ, u∗ T M ) − dim H∂0,1
¯ (Σ, u T M )
is given by the Riemann-Roch theorem:
index Du = index ∂¯u,∇ = 2n + 2c1 (A)).
3.4. TRANSVERSALITY
33
(See for example page 246 in Griffiths and Harris [25].)
We now return to the question of orientations. The determinant
det(D) = Λmax (ker D) ⊗ Λmax (ker D∗ )
of a Fredholm operator D : X → Y between complex Banach spaces carries a
natural orientation whenever the operator D is complex linear. In our case the
operator Du lies in a component of the space of Fredholm operators which contains
a complex linear operator and hence its determinant line det(Du ) = Λmax (ker Du )
also carries a natural orientation and this determines an orientation of Tu M(A, J) =
ker Du . Similar arguments were used by Donaldson in [11] for the orientation of
Yang-Mills moduli spaces and also by Ruan [64] in the present context. A slightly
different line of argument was used by McDuff in [42] where she established the
existence of a canonical stable almost complex structure2 on compact subsets of
the moduli space M(A, J).
2
Remark 3.3.6 Note that if J is integrable, then Du commutes with J, and so J
induces an (integrable) almost complex structure on M(A, J). This is, of course,
compatible with the orientation described above.
2
3.4
Transversality
The proofs of Theorems 3.1.2 and 3.1.3 are based on an infinite dimensional version
of Sard’s theorem which is due to Smale [80]. Roughly speaking, this says that if X
and Y are Banach manifolds and F : X → Y is a Fredholm map of index k then the
set Yreg of regular values of F is of the second category, provided that F is sufficently
differentiable (it should be at least C k+1 ). As in the finite dimensional case a point
y ∈ Y is called a regular value if the linearized operator DF(x) : Tx X → Ty Y is
surjective whenever F(x) = y. It then follows from the implicit function theorem
that F −1 (y) is a k-dimensional manifold for every y ∈ Yreg .
Our strategy is now to prove that the set of regular almost complex structures of
class C ` is generic with respect to the C ` -topology and then to take the intersection
of these sets over all `. This approach is due to Taubes [81]. We shall begin by
discussing the universal moduli space
M` (A, J ) = (u, J) ∈ X k,p × J ` | ∂¯J (u) = 0
of all J-holomorphic curves where J varies over the space J ` = J ` (M, ω) of all
almost complex structures of class C ` which are compatible with ω. Here ` ≥ 1
and, as above, X k,p denotes the space of W k,p -maps u : Σ → M with p > 2 and
1 ≤ k ≤ `. Recall from Proposition 3.2.2 that the space M` (A, J ) is independent
of the choice of k and p because every J-holomorphic curve is of class C ` whenever
J is of class C ` . The parameter space J ` is a smooth Banach manifold. Its tangent
2A
manifold X is said to have a stable almost complex structure if there is a number k such
that the Whitney sum T X ⊕ Rk of the tangent bundle T X with the trivial k-dimensional real
bundle carries an almost complex structure.
34
space TJ J ` at J consists of C ` -sections Y of the bundle End(T M, J, ω) whose fiber
at p ∈ M is the space of linear maps Y : Tp M → Tp M such that
Y J + JY = 0,
ω(Y v, w) + ω(v, Y w) = 0.
This space of C ` -sections is a Banach space and gives rise to a local model for the
space J ` via Y 7→ J exp(−JY ). The corresponding space of C ∞ -sections is not
a Banach space but only a Fréchet space. Our convention is that spaces with no
superscripts consist of elements which are C ∞ -smooth.
Proposition 3.4.1 For every class A ∈ H2 (M, Z) and every integer ` ≥ 1 the
universal moduli space M` (A, J ) is a smooth Banach manifold.
Proof: The map (u, J) 7→ ∂¯J (u) defines a section of the infinite dimensional vector
bundle E k−1,p → X k,p × J ` . Denote this section by
F : X k,p × J ` → E k−1,p ,
F(u, J) = ∂¯J (u).
The fiber of E k−1,p at (u, J) is the space
k−1,p
E(u,J)
= W k−1,p (Λ0,1 T ∗ Σ ⊗J u∗ T M )
of J-anti-linear 1-forms on Σ of class W k−1,p with values in the bundle u∗ T M → Σ.
We must prove that the differential
DF(u, J) : W k,p (u∗ T M ) × C ` (End(T M, J, ω))
→ W k−1,p (Λ0,1 T ∗ Σ ⊗J u∗ T M )
at a zero (u, J) is surjective whenever u is simple. This differential is given by the
formula
DF(u, J)(ξ, Y ) = Du ξ + 21 Y (u) ◦ du ◦ j
for ξ ∈ W k,p (u∗ T M ) and Y ∈ C ` (End(T M, J, ω)). The exact formula for Du is
not really relevant: all that matters is that it is a first order differential operator
which is Fredholm. We do however point out that the formula (3.2) in holomorphic
coordinates z = s + it on Σ and local coordinates on M specializes to
Du ξ = ζ ds − J(u)ζ dt,
ζ = ∂s ξ + J(u)∂t ξ + (∂ξ J(u))∂t u
whenever u is J-holomorphic. Recall from Proposition 3.2.2 that the J-holomorphic
curve u is in fact of class W `+1,p and hence, since p > 2, of class C ` . Hence the
coefficients of the first order terms in Du are of class C ` and those of the zero order
terms are of class C `−1 .
Since Du is Fredholm the operator DF(u, J) has a closed range and it suffices
to prove that this range is dense. We prove this first for k = 1. If the range is not
dense then, by the Hahn-Banach theorem, there exists a nonzero η ∈ Lq (Λ0,1 T ∗ Σ⊗J
u∗ T M ) with 1/p + 1/q = 1 which annihilates the range of DF(u, J). This means
that
Z
hη, Du ξi = 0
(3.6)
Σ
3.4. TRANSVERSALITY
35
for every ξ ∈ W 1,p (u∗ T M ) and
Z
hη, Y (u) ◦ du ◦ ji = 0
(3.7)
Σ
for every Y ∈ C ` (End(T M, J, ω)). The first equation asserts that the η is a weak
solution of Du∗ η = 0. Since the coefficients of the first order terms in Du are at least
C ` , the same is true for the adjoint operator Du∗ and it follows by elliptic regularity
that η is of class W `+1,r for any r > 0 and Du∗ η = 0. (Here the relevant elliptic
regularity statement is that if D is a first order elliptic operator with C ` -coefficients
and if Dη ∈ W k,p then η ∈ W k+1,p for k ≤ `.) Hence
0 = Du Du ∗ η = ∆η + lower order terms
and it follows from Aronszajn’s theorem 2.1.2 that if η vanishes on some open set
then η ≡ 0.
Since u is simple there exists, by Proposition 2.3.1, a point z0 ∈ Σ such that
du(z0 ) 6= 0,
u−1 (u(z0 )) = {z0 }.
We shall prove that η vanishes at z0 . Assume otherwise. Then it is easy to see that
there exists a Y0 ∈ End(Tu(z0 ) M, Ju(z0 ) , ωu(z0 ) ) such that
hη(z0 ), Y0 ◦ du(z0 ) ◦ j(z0 )i =
6 0.
Now use a smooth cutoff function to find a section Y ∈ C ` (End(T M, J, ω)) supported near u(z0 ) such that Yu(z0 ) = Y0 . For such a section the left hand side
of (3.7) does not vanish. This contradiction shows that η(z0 ) = 0. The same argument shows that η vanishes in a neighbourhood of z0 and hence η ≡ 0. This shows
that DF(u, J) has a dense range and is therefore onto in the case k = 1.3
To prove that DF(u, J) is onto for general k let η ∈ W k−1,p (Λ0,1 T ∗ Σ ⊗J
u∗ T M ) and choose, by surjectivity for k = 1, a pair ξ ∈ W 1,p (u∗ T M ) and
Y ∈ C ` (End(T M, J, ω)) such that
DF(u, J)(ξ, Y ) = η.
Then the above formula for DF(u, J) shows that
Du ξ = η − 21 Y (u) ◦ du ◦ j ∈ W k−1,p
and by elliptic regularity ξ ∈ W k,p (u∗ T M ). Hence DF(u, J) is onto for every
pair (u, J) ∈ M` (A, J ). Because Du is a Fredholm operator it is easy to prove
that the operator DF(u, J) has actually a right inverse. Hence it follows from
the infinite dimensional implicit function theorem that the space M` (A, J ) is an
infinite dimensional manifold.
2
3 In this argument we did not actually need Aronszajn’s theorem since we have proved directly
that η vanishes on the open and dense set of all injective points of u. However, in some situations
it is useful to keep J fixed on some subset of M and vary it in an open set which intersects the
image of every nonconstant J-holomorphic curve. Then our argument only shows that η vanishes
on some open set, and so we need Aronszajn’s theorem to be able to conclude that it vanishes
everywhere.
36
Proof of Theorem 3.1.2 (ii): Consider the projection
π : M` (A, J ) → J `
as a map between separable Banach manifolds. The tangent space
T(u,J) M` (A, J )
consists of all pairs (ξ, Y ) such that
Du ξ + 12 Y (u) ◦ du ◦ j = 0.
Moreover, the derivative
dπ(u, J) : T(u,J) M` (A, J ) → TJ J
is just the projection (ξ, Y ) 7→ Y . Hence the kernel of dπ(u, J) is isomorphic to the
kernel of Du . Moreover, its image consists of all Y such that Y (u) ◦ du ◦ j ∈ im Du .
This is a closed subspace of TJ J and, since DF(u, J) is onto, it has the same
(finite) codimension as the image of Du . It follows that dπ(u, J) is a Fredholm
operator and has the same index as Du . Moreover the operator dπ(u, J) is onto
precisely when Du is onto. Hence a regular value J of π is an almost complex
structure with the property that Du is onto for every simple J-holomorphic curve
u ∈ M` (u, J) = π −1 (J). In other words the set
`
Jreg
= J ∈ J ` | Du is onto for all u ∈ M` (u, J)
of regular almost complex structures is precisely the set of regular values of π. By
the Sard-Smale theorem, this set is of the second category in the sense of Baire
(a countable intersection of open and dense sets). Here we use the fact that the
manifold M` (A, J ) and the projection π are of class C `−1 . Hence we can apply the
Sard-Smale theorem whenever ` − 2 ≥ index Du = index π. Thus we have proved
`
is dense in J ` with respect to the C ` -topology. We shall now use
that the set Jreg
the following argument, which was suggested to us by Taubes [81], to deduce that
Jreg is of the second category in J with respect to the C ∞ -topology.
Consider the set
Jreg,K ⊂ J (M, ω)
of all smooth almost complex structures J ∈ J (M, ω) such that the operator Du
is onto for every J-holomorphic curve u : CP 1 → M which satisfies
kdukL∞ ≤ K
(3.8)
and for which there exists a point z ∈ CP 1 such that
inf
ζ6=z
d(u(z), u(ζ))
1
≥ .
d(z, ζ)
K
(3.9)
The latter condition guarantees that u is simple. Moreover, every simple J-holomorphic curve u : CP 1 → M satisfies these two conditions for some value of K > 0
and some point z ∈ CP 1 . Hence
\
Jreg =
Jreg,K .
K>0
3.4. TRANSVERSALITY
37
We shall prove that each set Jreg,K is open and dense in J (M, ω) with respect
to the C ∞ -topology. We first prove that Jreg,K is open or, equivalently, that its
complement is closed. Hence assume that the sequence Jν ∈
/ Jreg,K converges to
J ∈ J (M, ω) in the C ∞ -topology. Then there exists, for every ν, a sequence
of Jν -holomorphic curves uν : CP 1 → M which satisfy (3.8) and (3.9) for some
zν ∈ CP 1 such that the operator Duν is not surjective. It follows from standard elliptic bootstrapping arguments that there exists a subsequence uν 0 which converges,
uniformly with all derivatives, to a smooth J-holomorphic curve u : CP 1 → M (see
Appendix B). Choose the subsequence such that zν 0 converges to z. Then the limit
curve u satisfies the conditions (3.8) and (3.9) for this point z and, moreover, since
the operators Duν are all not surjective, it follows that Du cannot be surjective
either. This shows that J ∈
/ Jreg,K and thus we have proved that the complement
of Jreg,K is closed in the C ∞ -topology.
Now we prove that the set Jreg,K is dense in J (M, ω) with respect to the C ∞ topology. To see this note first that, by Remark 3.2.3,
`
Jreg,K = Jreg,K
∩J
`
⊂ J ` (M, ω) is the set of all C ` almost complex structures J ∈
where Jreg,K
J ` (M, ω) such that the operator Du is onto for every J-holomorphic curve u :
CP 1 → M of class C ` with with kdukL∞ ≤ K. Now the same argument as above
`
is open in J ` (M, ω) with respect to the C ` -topology. Moreshows that Jreg,K
`
`
`
over, Jreg ⊂ Jreg,K and hence, by the first part of the proof, Jreg,K
is also dense in
`
J (M, ω). This implies that Jreg,K is dense in J with respect to the C ` topology. To
`
see this let J ∈ J , approximate it in the C ` -topology by an element J 0 ∈ Jreg,K
, and
0
`
00
`
then approximate J in the C -topology by an element J ∈ Jreg,K ∩ J = Jreg,K .
Thus we have proved that the set Jreg,K is dense in J with respect to the C ` topology. But this holds for any ` and hence Jreg,K is dense in J with respect to
the C ∞ -topology. In fact, given J ∈ J choose a sequence Jν ∈ Jreg,K such that
kJ − Jν kC ν ≤ 2−ν . Then Jν converges to J in the C ∞ -topology. Thus Jreg is the
intersection of the countable number of open dense sets Jreg,K , and so has second
category as required.
2
This completes the proof of Theorem 3.1.2. Theorem 3.1.3 is proved in a similar
way: we take a path in the connected space J which joins J0 to J1 and then perturb
it to be transverse to π. This is possible by the Sard-Smale theorem quoted before.
Arguments analogous to those above (applied now to the space of paths) show that
the perturbed path can be chosen to be C ∞ -smooth.
Remark 3.4.2 The above proof shows that a point (u, J) ∈ M(A, J) is regular in
the sense of Definition 3.1.1 if and only if the operator
Du : W k,p (u∗ T M ) → W k−1,p (Λ0,1 T ∗ Σ ⊗J u∗ T M )
is surjective for some choice of k, p. Hence dπ(u, J) is surjective at this point. By
the implicit function theorem this implies that any smooth path [0, 1] → J : t 7→ Jt
which starts at J0 = J can be lifted, on some interval [0, ε), to a path [0, ε) →
M(A, J ) : t 7→ (ut , Jt ) in the universal moduli space which starts at u0 = u.
2
38
3.5
A regularity criterion
We close this chapter by establishing a simple criterion for an almost complex
structure J to be regular. In [27] Grothendieck proved that any holomorphic bundle
E over S 2 = CP 1 is holomorphically equivalent to a sum of holomorphic line
bundles. Moreover, this splitting is unique up to the order of the summands.
Hence E = L1 ⊕ · · · ⊕ Ln is completely characterized
by the set of Chern classes
P
c1 (L1 ), . . . , c1 (Ln ). Note that the sum c1 (E) = i c1 (Li ) is a topological invariant,
but that the set c1 (L1 ), . . . , c1 (Ln ) may vary as u : S 2 → M varies in a connected
component of the space of J-holomorphic A-spheres. In what follows we will identify
the Chern class c1 (L) with the corresponding Chern number hc1 (L), [S 2 ]i.
Lemma 3.5.1 Assume J is integrable and let u : CP 1 → M be a J-holomorphic
curve. Suppose that every summand of u∗ T M has Chern number c1 ≥ −1. Then
Du is onto.
Proof: If J is integrable then the operator Du is exactly the Dolbeault derivative
¯ and so it is surjective if and only if the Dolbeault cohomology group H 0,1
∂,
(u∗ T M )
∂¯
vanishes. But for any holomorphic line bundle L
1,0
1
∗ ∗
1
∼
H∂0,1
¯ (CP , L) = (H∂¯ (CP , L )) .
1
∗
But H∂1,0
¯ (CP , L ) is just the space of holomorphic 1-forms with values in the dual
bundle L∗ and so is isomorphic to the space H 0 (CP 1 , O(L∗ ⊗ K)) of holomorphic
sections of the bundle L∗ ⊗K where K = T ∗ CP 1 is the canonical bundle. This is an
easy special case of Kodaira-Serre duality (cf. [25, Ch 1 §2]) which can be checked
directly by considering the transition maps. But, by the Kodaira vanishing theorem,
a line bundle L0 on CP 1 has no holomorphic sections if and only if c1 (L0 ) < 0. Thus
Du is surjective if and only if c1 (L∗ ⊗ K) < 0. But c1 (L∗ ⊗ K) = −c1 (L) − 2 and
so this is equivalent to c1 (L) > −2.
2
Lemma 3.5.2 Let J be an integrable complex structure on a 4-dimensional manifold M and u : CP 1 → M be an embedded J-holomorphic curve. Then Du is onto
if and only if c1 (u∗ T M ) ≥ 1.
Proof: If u ∈ M(A, J) is an embedded J-holomorphic curve with image u(S 2 ) = C
then the pullback tangent bundle u∗ T M splits into complex subbundles
u∗ T M ∼
= TC M = T C ⊕ ν
where ν → C is a normal bundle with respect to some Hermitian structure. The
Chern number of T C is 2 and, since C is embedded, the Chern number of ν is
p = C · C. Hence
c1 (u∗ T M ) = 2 + C · C = 2 + p.
Now, according to Grothendieck, the bundle u∗ T M → S 2 splits into holomorphic subbundles
u∗ T M = L1 ⊕ L2
3.5. A REGULARITY CRITERION
39
whose Chern classes ki = c1 (Li ) satisfy
k1 + k2 = 2 + p.
Moreover, the tangent bundle T C of C is a holomorphic subbundle of L1 ⊕ L2 .
Therefore, it is either one of the summands, or it maps in a non-trivial way to both
of L1 and L2 . In the first case the other summand has Chern number p and so both
factors have Chern number ki ≥ −1. Hence it follows from Lemma 3.5.1 that the
pair (u, J) is regular. In the second case there is a non-trivial holomorphic section
of the line bundle Hom(T C, Li ) = T ∗ C ⊗ Li for i = 1, 2. This is possible if and only
if c1 (T ∗ C ⊗ Li ) = ki − 2 ≥ 0 for i = 1, 2 since line bundles with negative Chern
number have no non-zero sections. Hence it follows again from Lemma 3.5.1 that
the pair (u, J) is regular. Moreover, note that this second case can occur only when
p ≥ 2. This proves the lemma.
2
Remark 3.5.3 Let C = u(Σ) be the image of a J-holomorphic curve u : Σ → M in
a symplectic 4-manifold M . The adjunction formula of [44] asserts that the virtual
genus given by
2g(C) − 2 = C · C − hc1 , [C]i
is greater than or equal to the genus of Σ
g(C) ≥ g(Σ)
with equality if and only if C is embedded. In particular this implies that if a
homology class A ∈ H2 (M ) can be represented by an embedded J-holomorphic
curve u : Σ → M then every other J-holomorphic curve v : Σ → M which represents
A must also be embedded. In other words, one cannot deform an embedded Jholomorphic curve in a 4-manifold in such a way as to produce a singularity.
2
Corollary 3.5.4 Let J be an integrable complex structure on a 4-dimensional manifold M , and consider an embedded J-holomorphic sphere C with self-intersection
number C · C = p. Then J is regular for the class A = [C] if and only if p ≥ −1.
Proof: Lemma 3.5.2 and Remark 3.5.3.
2
Observe that this is the first place in our discussion where it is important to
restrict to spheres. Our other results apply to J-holomorphic curves with fixed
domain (Σ, j), but these are almost never regular because, even when J is integrable,
∗
the condition H∂0,1
¯ (Σ, u T M ) = 0 is usually not satisfied. The point is that when
J varies on M one must usually vary j on Σ in order to find a corresponding
deformation of the curve. Thus, one has to set up the Fredholm theory in such a
way that j is allowed to vary in Teichmüller space. This presents no problem at
this stage, but it does make the discussion of compactness more complicated, since
Teichmüller space, even when quotiented out by the mapping class group, is not
compact.
Example 3.5.5 Suppose that M is the product of S 2 with a Kähler manifold V .
Thus both ω = ω1 × ω2 and J = J1 × J2 respect the product structure. If A is
the homology class represented by the spheres S 2 × {pt} then the J-holomorphic
A-spheres are given by the maps z 7→ (φ(z), x0 ) where φ is a fractional linear
transformation and x0 ∈ V . Hence the above lemma implies that J ∈ Jreg (A). 2
40
So far, there are no general methods known for proving that a given nonintegrable J is regular for the class A. However, in the case of 4-manifolds one
can do this by using positivity of intersections provided that c1 (A) > 0: see Hofer–
Lizan–Sikorav [31] and Lorek [40]. When extending these results to curves of higher
genus one has to be somewhat careful because the question of whether a line bundle
has holomorphic sections is no longer purely topological, depending on the Chern
number alone. Other respects in which moduli spaces of curves of higher genus
differ from those of spheres are discussed in [47].
Chapter 4
Compactness
Because any manifold V is cobordant to the empty manifold via the non-compact
cobordism V × [0, 1), Theorem 3.1.3 is useless unless one can establish some kind
of compactness. Now in the case Σ = S 2 the manifold M(A, J) itself cannot be
compact (unless it is empty) since the non-compact group G = PSL(2, C) of biholomorphic maps of S 2 acts on this space by reparametrization. However, in some
cases the space M(A, J)/G of unparametrized spheres is compact.
If J is tamed by a symplectic form ω then it follows from the energy identity
(see Lemma 4.1.2 below) that there is a uniform bound on the W 1,2 -norm of all
J-holomorphic curves in a given homology class. This is the Sobolev borderline case
kp = 2 and, as a result, the space of such curves will in general not be compact.
This is due to the conformal invariance of the energy in 2 dimensions and leads to
the phenomenon of bubbling, which was first discovered by Sacks and Uhlenbeck
in the context of minimal surfaces [71]. In fact, it follows from the usual elliptic
bootstrapping argument that any sequence uν in M(A, J) which is bounded in the
W 1,p -norm for some p > 2 has a subsequence which converges uniformly with all
derivatives. On the other hand if the first derivatives of uν are only bounded in L2
but not in Lp a simple geometric argument using conformal rescaling allows one
to construct a J-holomorphic map v : C → M with finite area which, by “removal
of singularities”, can be extended to S 2 = C ∪ {∞}. This is the phenomenon of
“bubbling off of spheres”. Sometimes, by choosing the class A carefully, one can
show that this cannot happen. In this case, the space M(A, J)/G is compact. In
other cases, bubbling off can occur, and one must proceed with more care.
The method of proof used below is very close to that followed by Floer. Gromov’s original approach was somewhat different. He argued geometrically, using
isoperimetric inequalities and the Schwartz Lemma for conformal maps. More details of his proofs have been written up by Pansu [61]. The flavour of his arguments
may be sampled in the proof of Theorem 4.2.1 (removal of singularities) below.
Other accounts of this subject can be found in [32], [62], [73], and [88]. In Section 4.3 we shall prove the simplest version of the compactness theorem. This
gives a criterion for the moduli space of unparametrized curves to be compact.
The criterion is that A is minimal in the sense that there is no other class B with
0 < ω(B) < ω(A). In particular this implies that A is not a multiple class so that
the space M(A, J) contains all J-holomorphic A-curves. If this is not satisfied,
41
42
CHAPTER 4. COMPACTNESS
one can form a compactification by adding suitable “cusp-curves” and this will be
discussed in detail in Sections 4.4 and 4.5. The terminology cusp-curve was introduced by Gromov. However, these curves would be called reducible curves in the
setting of algebraic geometry.
We will assume, for simplicity, that the Riemann surface Σ is a sphere. In fact,
the arguments go through without essential change to the general case provided
that one fixes the complex structure j on Σ. Note that in all cases, the bubbles
which appear are spherical. We will denote the reparametrization group by G.
Thus G is the non-compact group PSL(2, C) acting on S 2 = C ∪ {∞} by fractional
linear transformations
az + b
a b
,
A=
∈ SL(2, C).
(4.1)
φA (z) =
c d
cz + d
We begin with a discussion of metrics on M which are related to J and ω. We
will assume that J : T M → T M is an ω-tame almost complex structure on M .
This means that
ω(v, Jv) > 0
for
v 6= 0.
This condition is weaker than compatibility since we no longer require the bilinear
form (v, w) 7→ ω(v, Jw) to be symmetric. However there is an induced Riemannian
metric
gJ (v, w) = hv, wiJ = 21 (ω(v, Jw) + ω(w, Jv)) .
Henceforth, we assume that M is provided with this metric. It is easy to check
that the endomorphism J is an isometry and is skew self-adjoint with respect to
this metric. It follows that a J-holomorphic map u is conformal with respect to the
Poincaré metric on (Σ, j) and the above metric on M .
4.1
Energy
We define the energy of a smooth map u : Σ → M to be the L2 -norm of the 1-form
du ∈ Ω1 (u∗ T M ):
Z
1
E(u) =
|du|2J dA.
2 Σ
Here the norm |du|J is not well defined unless we fix a metric on Σ. However, the
following exercise shows that the product |du|2J dA is independent of the choice of
this metric.
Exercise 4.1.1 Let Σ be a Riemann surface with complex structure j, E → Σ
be a Riemannian vector bundle, and ζ ∈ Ω1 (E) be a 1-form. Choose a conformal
coordinate chart α : Uα → C on Σ and an orthogonal trivialization and denote
by z = s + it = α(p) the coordinates of a generic point p ∈ Uα ⊂ Σ. Moreover,
choose a local orthogonal trivialization φα : π −1 (Uα ) → α(Uα ) × Rn of E such that
p1 ◦ φα = α ◦ π. In these coordinates the 1-form ζ can be represented by the form
ζ α = ξ α ds + η α dt ∈ Ω1 (α(Uα ), Rn )
4.2. REMOVAL OF SINGULARITIES
43
where p2 ◦ φα ◦ ζ = α∗ ζ α . Prove that the local expressions
|ζ|2 dA := α∗ |ξ α |2 + |η α |2 ds ∧ dt
on Uα determine a well defined global 2-form on Σ. Its integral is the L2 -norm of
ζ.
2
Lemma 4.1.2 If J is ω-tame then the energy identity
Z
E(u) = u∗ ω
(4.2)
Σ
holds for all J-holomorphic curves u.
2
Proof: Exercise.
Next we give a precise statement of the basic compactness result which is proved
by “elliptic bootstrapping”. In this theorem the domain Σ may be noncompact.
The proof will be carried out in Appendix B.
Theorem 4.1.3 Let uν : Σ → M be a sequence of J-holomorphic curves such that
sup kduν kL∞ (K) < ∞
ν
for every compact subset K ⊂ Σ. Then uν has a subsequence which converges
uniformly with all derivatives on compact subsets of Σ.
The conclusion of the previous theorem continuous to hold when the maps uν
are uniformly bounded in the W 1,p -norm for some p > 2. Now, if the elements
of the sequence uν all represent the same homology class then the energy identity
of the Lemma 4.1.2 guarantees a uniform bound for the W 1,2 -norm. However, the
compactness theorem breaks down in the case p = 2 and the conformal invariance of
the energy leads to the phenomenon of bubbling. An important ingredient in understanding this phenomenon is the removable singularity theorem for J-holomorphic
curves of finite energy which we will discuss next.
4.2
Removal of Singularities
The removable singularity theorem says that any J-holomorphic curve u : B −{0}→
M on the punctured disc which has finite energy extends smoothly to the disc.
Theorem 4.2.1 (Removal of singularities) Let J be a smooth ω-tame almost
complex structure on a compact manifold M with associated metric gJ . If u :
B − {0} → M is a J-holomorphic curve with finite energy E(u) < ∞ then u
extends to a smooth map B → M .
44
Proof of continuity: Here we follow essentially the line of argument in Gromov’s
original work. We assume for simplicity that J is compatible with ω so that the Jholomorphic curves minimize the energy. As pointed out by Pansu in [61, §37], one
can prove the removable singularity theorem in this case by using the monotonicity
theorem for minimal surfaces. This states that there are constants c > 0 and ε0 > 0
(which depend on M and the metric gJ ) such that for every minimal surface S in
(M, gJ ) which goes through the point x
areagJ (S ∩ B(x, )) ≥ c2
for 0 < ε < 0 . (See [37, 3.15].) To apply this, suppose that u(z) has two limit
points p and q as z → 0. If δ is chosen to be less than d(p, q)/3, then the monotonicity theorem implies that each connected component of u−1 (B(p, δ)) which meets
u−1 (B(p, δ/2)) is taken by u to a surface in M which has area ≥ cδ 2 /4. Therefore,
because Im u is minimal and has finite area (or energy) E(u) by (4.2), there can only
be a finite number of such components. Similar remarks apply to q. Hence there
exists an r0 > 0 such that, for any r < r0 , the image γr of the circle {z ∈ C | |z| = r}
under u meets both B(p, δ/2) and B(q, δ/2), and so must have length `(γr ) > δ.
But then, the conformality of u implies that |du| = 1r |∂u/∂θ|, and we find that
|∂u/∂θ|2
rdr ∧ dθ
r2
(0,1]×S 1
Z
2
Z
1
≥
|∂u/∂θ|dθ
dr
2πr
1
(0,1]
S
Z
`(γr )2
dr
=
(0,1] 2πr
Z
δ2
≥
dr,
(0,r0 ] 2πr
Z
E(u)
=
which is impossible because E(u) is finite. The geometric idea here is that, because
of the conformality, if the loops γr are long they must also stretch out in the radial
direction, and hence form a surface of infinite area.
2
We now give a proof of the smoothness of the extension which relies on the
following a priori estimate.
Lemma 4.2.2 (A priori estimate) Assume M is compact and J is a smooth ωtame almost complex structure. Then there exists a constant ~ > 0 such that the
following holds. If r > 0 and u : Br → M is a J-holomorphic curve such that
Z
E(u; Br ) =
|du|2 < ~
Br
then
8
|du(0)| ≤ 2
πr
2
Z
|du|2 .
Br
The proof relies on a partial differential inequality of the form ∆e ≥ −Ae2 for
the energy density e = |du|2 . The details are carried out in [73] for example. As a
4.2. REMOVAL OF SINGULARITIES
45
consequence of this estimate we obtain the following isoperimetric inequality which
plays a crucial role in the proof of Theorem 4.2.1 and is related to the monotonicity
property of minimal surfaces used above.
Lemma 4.2.3 (Isoperimetric inequality) Assume that M is compact and that
J is a smooth ω-tame almost complex structure. Let u : B − {0} → M be a
J-holomorphic curve on the punctured disc with finite energy. Then there exist
constants δ > 0 and c > 0 such that
E(u; Br ) ≤ c `(γr )2 ,
0<r<δ
where γr (θ) = u(reiθ ) and `(γr ) denotes the length of the loop γr .
Proof: In view of Lemma 4.2.2 we have
|du(reiθ )|2 ≤
8
ε(2r),
πr2
ε(r) = EBr (u)
for r ≤ δ. Now the derivative of γr has norm |γ̇r (θ)| = r du(reiθ ) . Hence the
length of the loop γr is given by
2π
Z
p
du(reiθ ) dθ ≤ 32πε(2r).
`(γr ) = r
0
Hence `(γr ) converges to zero with r.
Now every sufficiently short loop γ : S 1 → M has a unique local extension uγ :
B → M defined by uγ (reiθ ) = expγ(0) (rξ(θ)) where ξ(θ) ∈ Tγ(0) M is determined by
the condition expγ(0) (ξ(θ)) = γ(θ). The area of this disc is the local symplectic
action of γ and is bounded by the square of the length of γ. In other words, there
exists a constant c > 0 such that for every sufficiently short loop γ : S 1 → M
Z
a(γ) = u∗γ ω ≤ c `(γ)2
Denote by ur = uγr : B1 → M the extension of the loop γr and consider the sphere
vρr : S 2 → M obtained from u|Br −Bρ with the boundary circles γρ and γr filled
in by the discs uρ and ur . This sphere is contractible. It is the restriction of the
smooth map B1 × [ρ, r] → M : (z, τ ) 7→ uτ (z) to the boundary of the cylinder.
Hence
Z
Z
u∗ρ ω =
E(u; Br − Bρ ) +
B1
u∗r ω
B1
for 0 < ρ < r for r sufficiently
small. Take the limit ρ → 0 to obtain the required
R
inequality E(u; Br ) = u∗r ω ≤ c `(γr )2 .
2
Proof of Theorem 4.2.1: Continue the notation of Lemma 4.2.3. In particular
Z
2
Z
|du| =
ε(r) = E(u; Br ) =
Br
r
Z
ρ
0
0
2π
du(ρeiθ )2 dρdθ.
46
It follows from the isoperimetric inequality of Lemma 4.2.3 that for r sufficiently
small
ε(r) ≤
=
c `(γr )2
Z 2π
2
du(reiθ ) dθ
cr2
0
2π
Z
≤
2πcr2
=
2πcrε̇(r).
du(reiθ )2 dθ
0
With µ = 1/2πc this can be rewritten as µ/r ≤ ε̇(r)/ε(r). Integrating this inequality
µ
from r to r1 we obtain (r1 /r) ≤ ε(r1 )/ε(r) and hence
ε(r) ≤ c1 rµ
where c1 = r1−µ ε(r1 ). For ρ sufficiently small the estimate of Lemma 4.2.2 shows
that
8
c2
|du(ρeiθ )|2 ≤
ε(2ρ) ≤ 2−µ
(4.3)
2
πρ
ρ
with a suitable constant c2 > 0 which is independent of ρ and θ. With 2 < p <
4/(2 − µ) this implies
Z
p
|du|
Br
Z
r
Z
=
0
Z
≤ c3
2π
ρ|du(ρeiθ )|p dθdρ
0
r
ρ1−p(1−µ/2) dρ.
0
Since p < 4/(2 − µ) we have 1 − p(1 − µ/2) > −1 and hence the integral is finite.
Hence u ∈ W 1,p with p > 2. Moreover, it follows from (4.3) that u is Hölder
continuous. Since M is compact this implies (again) that u extends to a continuous
map on the closed unit disc. Now it follows from elliptic regularity in a local
coordinate chart that u is smooth on the closed unit disc (see Proposition 3.2.2).
2
This completes the proof of the removable singularity theorem. A similar result
is true for boundary points of J-holomorphic curves: see Oh [59].
Exercise 4.2.4 Prove directly from (4.3) that u is Hölder continuous with exponent µ/2. This is needed to prove that u(Br − {0}) lies in a single coordinate chart
for r > 0 sufficiently small.
2
4.3
Bubbling
The next theorem shows how bubbles appear. We shall begin by discussing some notation. The most convenient way of describing holomorphic spheres is to identify S 2
with CP 1 = C∪{∞}. If S 2 denotes the unit sphere in R3 then such an identification
is given by stereographic projection. Different choices of a stereographic projection
4.3. BUBBLING
47
correspond to the action of SO(3) ' SU(2)/{±1l} ⊂ PSL(2, C) on CP 1 = C ∪ {∞}.
With this identification a J-holomorphic sphere u ∈ M(A, J) is a smooth Jholomorphic curve u : C → M such that the map C − {0} → M : z 7→ u(1/z)
extends to a smooth map on C. This class of maps is preserved under composition
with fractional linear transformations φA : C ∪ {∞} → C ∪ {∞}. A sequence of
J-holomorphic curves uν : C → M is said to converge on C ∪ {∞} if both uν (z)
and uν (1/z) converge uniformly with all derivatives on compact subsets of C. We
make use of the following estimate on the derivatives.
Lemma 4.3.1 If u : C → M represents a smooth map S 2 → M then there exists
a constant c > 0 such that
c
|du(z)| ≤
1 + |z|2
for z ∈ C.
Proof: Define v(z) = u(1/z) and note that the map φ(z) = 1/z satisfies |dφ(z)| =
|z|−2 . Hence |du(z)| = |z|−2 |dv(1/z)| and this proves the estimate for |z| ≥ 1 with
c = kdvkL∞ (B1 ) . For |z| ≤ 1 the estimate obviously holds with c = kdukL∞ (B1 ) . 2
Theorem 4.3.2 Assume that there is no spherical homology class1 B ∈ H2 (M )
such that 0 < ω(B) < ω(A). Then the moduli space M(A, J)/G is compact.
Proof: Let uν : C → M be a sequence of J-holomorphic curves which represent
the class A. We must prove that there exists a sequence Aν ∈ PSL(2, C) such that
uν ◦ φAν has a subsequence which converges on C ∪ {∞}.
It follows from Lemma 4.3.1 that |duν (z)| assumes its maximum at some point
aν ∈ C. Denote
cν = |duν (aν )| = kduν kL∞
and define the reparametrized curve vν : C → M by
vν (z) = uν (aν + c−1
ν z).
This curve satisfies
|dvν (0)| = 1,
kdvν kL∞ ≤ 1,
E(vν ) = E(uν ) = ω(A).
The last identity follows from the conformal invariance of the energy and the formula (4.2). By Theorem 4.1.3 there exists a subsequence, still denoted by vν ,
which converges uniformly with all derivatives on compact sets. The limit function
v : C → M is again a J-holomorphic curve such that
Z
|dv(0)| = 1,
0 < E(v) =
v ∗ ω ≤ ω(A).
C
Now the removable singularity theorem implies that the map C − {0} → M : z 7→
v(1/z) extends to a smooth map on C and hence v represents a J-holomorphic
sphere.
1 A homology class B ∈ H (M ) is called spherical if it is in the image of the Hurewicz
2
homomorphism π2 (M ) → H2 (M ).
48
In order to show that M(A, J)/G is compact, we must prove that the functions
vν0 (z) = vν (1/z) converge to v(1/z) uniformly on compact neighbourhoods of 0.
Assume otherwise that
c0ν = kdvν0 kL∞ → ∞,
passing to a subsequence if necessary. As before, let a0ν ∈ C be the point at
which |dvν0 (z)| attains its maximum. Since vν0 (z) converges to v(1/z) uniformly on
compact subsets of C − {0} it follows that a0ν → 0. Consider the reparametrized
curves
−1
wν (z) = vν0 (a0ν + c0ν z).
These maps again satisfy
|dwν (0)| = 1,
kdwν kL∞ ≤ 1,
E(wν ) = ω(A).
Hence, passing to a further subsequence, we may assume that wν converges uniformly with all derivatives on compact sets to a J-holomorphic curve w : C → M
such that
Z
|dw(0)| = 1,
0 < E(w) =
w∗ ω ≤ ω(A).
C
Again it follows from the removable singularity theorem that w(1/z) extends to a
smooth J-holomorphic curve at 0 and hence represents a J-holomorphic sphere.
Now let B and C denote the homology classes represented by v and w, respectively. Then ω(B) and ω(C) are positive and we derive the desired contradiction
by showing that
ω(B) + ω(C) ≤ ω(A).
This holds because v and w are limits of “disjoint pieces” of the sequence uν . More
precisely, denote by BR = BR (0) ⊂ C denotes the ball of radius R centered at 0
and by
Z
w∗ ω
E(w; Ω) =
Ω
the energy of the J-holomorphic curve w on the domain Ω ⊂ C. Then for every
ε > 0 we have
ω(C)
=
=
=
≤
=
=
lim E(w; BR )
R→∞
lim lim E(wν ; BR )
R→∞ ν→∞
lim lim E(vν0 ; BRc0ν −1 (a0ν ))
R→∞ ν→∞
lim lim E(vν0 ; Bε )
R→∞ ν→∞
lim (E(vν ) − E(vν ; B1/ε ))
ν→∞
ω(A) − E(v; B1/ε ).
Take the limit ε → 0 to obtain the required inequality ω(C) ≤ ω(A) − ω(B). Note
that ω(B) + ω(C) need not equal ω(A) because there may be yet other bubbles
which we have not detected. Different choices of the center and size of the rescaling
may give rise to different limits.
2
A simple modification of the above proof with varying almost complex structures
gives the following result.
4.3. BUBBLING
49
Corollary 4.3.3 Assume that there is no spherical homology class B ∈ H2 (M )
such that 0 < ω(B) < ω(A). Assume that Jν is a sequence of almost complex
structures which converges to J in the C ∞ -topology. Let uν : C ∪ {∞} → M be
a sequence of Jν -holomorphic A-spheres. Then there exist matrices Aν ∈ SL(2, C)
such that uν ◦ φAν has a convergent subsequence.
We say that the class A is indecomposable if it does not decompose as a sum
A = A1 + · · · + Ak of classes which are spherical and satisfy ω(Ai ) > 0 for all i.
Note that A may be an indecomposable class without satisfying the assumption of
Theorem 4.3.2. For indecomposable classes the conclusion of Theorem 4.3.2 remains
valid but the proof requires the more sophisticated compactness theorem discussed
in the next section. Here we only state the result.
Theorem 4.3.4 If A is indecomposable then the moduli space M(A, J)/G of unparametrized J-holomorphic A-spheres is compact for all ω-compatible J.
Example 4.3.5 Let M = S 2 × S 2 with symplectic form ωλ = λω1 × ω2 where
each ωi is an area form on the sphere with total area π and where λ ≥ 1. Let
A = [S 2 × pt] and B = [pt × S 2 ]. If λ = 1, then ωλ (A) = π is the smallest
positive value taken by [ωλ ] on π2 (M ). Thus A is indecomposable and M(A, J)/G
is compact. But if λ > 1, ωλ is positive on the class A − B of the anti-diagonal
{(z, α(z)) : z ∈ S 2 } (where α is the antipodal map), and it becomes possible for the
set of J-holomorphic A-spheres to be non-compact. If J is a product, M(A, J)/G
is compact, but there are ωλ -compatible J for which it is not. For example, one
can take J to be Hirzebruch’s complex structure on S 2 × S 2 which is obtained by
identifying S 2 × S 2 with the projectivization of the rank 2 complex vector bundle
over S 2 = CP 1 which has first Chern class equal to 2. However, one can show that,
no matter what λ is, the moduli space M(A, J)/G is compact for a generic J. 2
Exercise 4.3.6 Let M = S 2 × T2n−2 be the product of the 2-sphere with a torus
with symplectic form ω = ω1 × ω2 , and let A = [S 2 × {pt}]. Show that A is
indecomposable.
2
When one applies these results to get information about the symplectic manifold, the most useful tool is the evaluation map. If the curves under consideration
are spheres, then the domain of this map is the quotient space M(A, J)×G S 2 where
the reparametrization group G acts on the product diagonally
φ · (u, z) = (u ◦ φ−1 , φ(z)).
The evaluation map e = eA = eA,J : M(A, J) ×G S 2 → M is given by the formula
e(u, z) = u(z).
Theorem 4.3.7 Let A be a indecomposable class, and J1 , J2 two elements of Jreg .
Then the evaluation maps eA,J1 and eA,J2 are compactly bordant.
Proof: The evaluation maps extend over M(A, {Jλ }λ ) ×G S 2 .
2
50
4.4
Gromov compactness
Throughout we assume that (M, ω) is a compact symplectic manifold and denote
by J = Jτ (M, ω) the space of smooth ω-tame almost complex structures on M . If
J ∈ J and A ∈ H2 (M, Z) is not an indecomposable class then the unparametrized
moduli space C(A, J) = M(A, J)/G will in general not be compact. A sequence of
A-curves may degenerate into a cusp-curve C. This is a connected union
C = C1 ∪ C2 ∪ · · · ∪ CN
of J-holomorphic spheres C j which are called components. Each component is
parametrized by a smooth nonconstant J-holomorphic map uj : CP 1 → M which
is not required to be simple. A special case occurs when N = 1 and the curve u = u1
is multiply covered. Then the convergence may be uniform, with all derivatives, but
the limit-curve u is not included in the space M(A, J) which only consists of simple
curves. In [64] Ruan uses the term reducible curve for a collection u = (u1 , . . . , uN )
of nonconstant J-holomorphic curves with a connected image such that either N ≥
2 or N = 1 and u = u1 is multiply covered. We shall adopt Gromov’s terminology
in [26] and use the term cusp-curve.
Remark 4.4.1 A cusp-curve u = (u1 , . . . , uN ) can be ordered such that the set
C 1 ∪ · · · ∪ C j with C k = uk (CP 1 ) is connected for every j. This means that there
exist numbers
j2 , . . . , j N ,
1 ≤ jk < k,
and points wk , zk ∈ CP 1 such that
ujk (wk ) = uk (zk ).
As a matter of normalization we may assume for example that zk = ∞ for every k.
2
Remark 4.4.2 A cusp-curve u = (u1 , . . . , uN ) can be parametrized by a single
smooth but not J-holomorphic map v : CP 1 → M . To see this order the curves
u1 , . . . , uN as in the previous remark and argue by induction as follows. Assume
that v j : CP 1 → M has been constructed for j ≥ 1 as to parametrize C 1 ∪ · · · ∪ C j .
Assume without loss of generality that v j (∞) = uj+1 (0) and choose a smooth map
v j+1 : C ∪ {∞} → M which covers v j (CP 1 ) on the unit disc, maps the unit circle
to v j (∞) = uj+1 (0) and covers C j+1 on the complement of the unit disc. More
explicitly, choose a smooth cutoff function β : R → [0, 1] such that β(r) = 1 for
r ≤ 1/2 and r ≥ 2 and β(r) = 0 for r close to 1, and define
j
v (β(|z|2 )−1 z), if |z| < 1,
v j+1 (z) =
uj+1 (β(|z|2 )z), if |z| > 1.
This curve parametrizes C 1 ∪ · · · ∪ C j+1 .
2
A sequence of J-holomorphic curves uν : CP 1 = C ∪ {∞} → M is said to
converge weakly to a curve u = (u1 , . . . , uN ) (which may consist only of one
component) if the following holds.
4.4. GROMOV COMPACTNESS
51
(i) For every j ≤ N there exists a sequence φjν : CP 1 → CP 1 of fractional linear
transformations and a finite set X j ⊂ CP 1 such that uν ◦ φjν converges to uj
uniformly with all derivatives on compact subsets of CP 1 − X j .
(ii) There exists a sequence of orientation preserving (but not holomorphic) diffeomorphisms fν : CP 1 → CP 1 such that uν ◦ fν converges in the C 0 -topology
to a parametrization v : CP 1 → M of the cusp-curve u as in Remark 4.4.2.
S
It follows from this definition that every point on the union C = j uj (CP 1 ) is a
limit of some sequence pν ∈ uν (CP 1 ). It follows also from the definition of weak
convergence that the map uν : CP 1 → M is homotopic to the connected sum
u1 #u2 # · · · #uN : CP 1 → M
for ν sufficiently large. Hence, in particular,
ω(Aν ) =
N
X
j=1
ω(Aj ),
c1 (Aν ) =
N
X
c1 (Aj ),
j=1
where Aν ∈ H2 (M, Z) is the homology class of uν and Aj is the homology class of
uj . In our applications below we shall in general assume that all the uν represent the
same homology class A ∈ H2 (M, Z) and it then follows from weak convergence that
the connected sum u1 # · · · #uN also represents the class A. We point out that the
notion of weak convergence also makes sense when the uν are not J-holomorphic.
Theorem 4.4.3 (Gromov’s compactness) Assume M is compact and let Jν ∈
Jτ (M, ω) be a sequence of ω-tame almost complex structures which converges to J
in the C ∞ -topology. Then any sequence uν : CP 1 → M of Jν -holomorphic spheres
with supν E(uν ) < ∞ has a subsequence which converges weakly to a (possibly
reducible) J-holomorphic curve u = (u1 , . . . , uN ).
Various versions of this theorem are proved in [32], [61], [62], and [88]. The
proof given in the next section essentially follows the line of argument in [32].
Corollary 4.4.4 Given K > 0, every element J ∈ J has an open neighbourhood
N (J) such that there are only finitely many classes A with ω(A) ≤ K which have
J 0 -holomorphic representatives for some J 0 ∈ N (J).
Proof: If not, there would be a sequence of curves, each in a different homology
class, of bounded energy. But these homology classes lie in the discrete lattice
H2 (M ; Z), and so can have no convergent subsequence.
2
If uν is a weakly converging sequence, we will see in Lemma 4.5.5 that there is
a finite set X such that uν converges to some J-holomorphic map u∞ on compact
subsets of CP 1 − X. The corresponding component of the limiting cusp-curve can
be thought of as basic one, the other components being “bubbles ”at the points of
X. When Σ is the sphere, this remark has little force, but if one considers a limit
of maps uν whose domain is a fixed Riemann surface Σ, the basic component is
the only one which can have genus > 0. Because there is no obvious holomorphic
parametrization of the different components of a limiting cusp-curve, it is hard to
52
define a good compactification for spaces M of parametrized curves. However, all
that is needed to define symplectic invariants is to compactify the space C(A, J) =
M(A, J)/G of unparametrized curves. We shall describe such a compactification in
the next chapter. This compactification can be understood without going through
the detailed analysis involved in the proof of Gromov’s compactness theorem and
the reader may wish to go directly to Chapter 5.
4.5
Proof of Gromov compactness
One key step in the proof is the following result about the energy of a J-holomorphic
curve on an arbitrarily long cylinder. It asserts that if the energy is sufficiently small
then it cannot be spread out uniformly but must be concentrated near the ends.
We phrase the result in terms of closed annuli A(r, R) = BR − int Br for r < R.
The relative size of the radii r, R will be all-important in what follows. One should
think of the ratio R/r as being very large, and much bigger than eT .
Lemma 4.5.1 Let (M, ω) be a compact symplectic manifold and J be an ω-tame
almost complex structure. Then there exist constants c > 0, ~ > 0, and T0 > 0
such that the following holds. If u : A(r, R) → M is a J-holomorphic curve such
that E(u) < ~ then
c
E(u; A(eT r, e−T R)) ≤ E(u; r, R)
T
and
r
Z 2π
E(u; A(r, R))
T +iθ
−T +iθ
dist u(re
), u(Re
) dθ ≤ c
T
0
for T ≥ T0 .
Proof: Choose the constant ~ > 0 as in Lemma 4.2.2 and consider the Jholomorphic curve v(τ + iθ) = u(eτ +iθ ) for log r < τ < log R and θ ∈ S 1 = R/2πZ.
Then for log r + T < τ < log R − T , the projection BT → [log r, log R] × S 1 is at
most a T -fold cover and so we have E(v; BT (τ + iθ)) ≤ T E(v) = T E(u). Hence,
by Lemma 4.2.2,
2
|dv(τ + iθ)| ≤
8E(u)
,
πT
log r + T < τ < log R − T.
If T is sufficiently large then the loop γτ (θ) = v(τ + iθ) is sufficiently short and
therefore has a local symplectic action a(γτ ) as in the proof of Lemma 4.2.3. Recall
that this action is defined as the integral of ω over a local disc uτ : B1 → M with
uτ (eiθ ) = γτ (θ) and satisfies
|a(γτ )| ≤ c1 `(γτ )2 ≤ c2
E(u)
T
for log r + T < τ < log R − T . Since u is a J-holomorphic curve we have
d
a(γτ ) =
dτ
Z
0
2π
kγ̇τ k 2
k∂τ γτ kL2 p
|γ̇τ (θ)|2 dθ ≥ √ L `(γτ ) ≥ √
|a(γτ )|.
2πc1
2π
4.5. PROOF OF GROMOV COMPACTNESS
53
In particular, the function τ 7→ a(γτ ) is strictly increasing. If a(γτ ) > 0 then
dp
a(γτ ) ≥ c−1
3 k∂τ γτ kL2
dτ
and a similar inequality holds when a(γτ ) < 0. Integrating these from τ0 = log r+T
to τ1 = log R − T (after splitting this interval into two according to the sign of
a(γτ ) if necessary) we obtain
r
Z τ1
p
p
E(u)
|a(γτ0 )| + |a(γτ1 )| ≤ c4
.
k∂τ γτ kL2 dτ ≤ c3
T
τ0
p
Since k∂τ γτ kL2 ≤ c5 E(u)/T for τ0 ≤ τ ≤ τ1 this implies
Z τ1
E(u)
2
k∂τ γτ kL2 dτ ≤ c6
.
E(u; A(eT r, e−T R)) =
T
τ0
Moreover,
Z
2π
Z
dist(γτ0 (θ), γτ1 (θ)) dθ
τ1
2π
Z
≤
0
|∂τ γτ | dθdτ
τ0
≤
≤
√
0
Z
2π
r
c7
τ1
τ0
k∂τ γτ kL2 dτ
E(u)
T
2
and this proves the lemma.
1
It is convenient to introduce some notation. Let uν : CP → M be any sequence
of smooth maps. A point z ∈ CP 1 is called regular for uν if there exists an ε > 0
such that the sequence duν is uniformly bounded on Bε (z). A point z ∈ Σ is called
singular for uν if it is not regular. This means that there exists a sequence zν → z
such that |duν (zν )| is unbounded. A singular point z for uν is called rigid if it is
singular for every subsequence of uν . This means that the sequence zν → z can
be chosen such that |duν (zν )| → ∞.2 It is called tame if it is isolated (no other
singular points in some neighbourhood of z) and the limit
Z
mε (z) = lim
u∗ν ω
ν→∞
Bε (z)
exists and is finite for every sufficiently small ε > 0. In this case the mass of the
singularity is defined to be the number
m(z) = lim mε (z).
ε→0
This limit exists because the function ε 7→ mε (z) is non-decreasing. We shall
prove that for a sequence of J-holomorphic curves each tame singularity has mass
m(z) ≥ ~ where ~ > 0 is defined by the following lemma.
2 Warning: If you identify CP 1 with C ∪ {∞} then the point infinity is regular iff it is regular
for the sequence z 7→ uν (1/z). Similarly for singular and rigid. In particular, ∞ is a rigid singular
point if there exists a sequence zν → ∞ such that |duν (zν )|·|zν |2 → ∞ where the norm of duν (zν )
is taken with respect to the standard metric on C.
54
Lemma 4.5.2 For every compact symplectic manifold (M, ω) and every almost
complex structure J ∈ Jτ (M, ω) there exists a constant ~ > 0 such that
E(u) > ~
for every nonconstant J-holomorphic curve u : CP 1 → M .
Proof: Choose ~ > 0 as in Lemma 4.2.2. Let u : C ∪ {∞} → M be a Jholomorphic curve with E(u) ≤ ~. Then for every z ∈ C and every r > 0 we have
|du(z)|2 ≤ 8E(u, Br (z))/πr2 ≤ 8~/πr2 . Hence du(z) = 0 for every z and therefore
u is constant.
2
We shall need the the following observation about complete metric spaces which
is due to Hofer. The proof is left as an exercise.
Lemma 4.5.3 Let X be a metric space and f : X → R be continuous and nonnegative. Given ζ ∈ X and δ > 0 assume that the closed ball Bδ (ζ) = {x ∈
X | dist(x, ζ) ≤ δ} is complete. Then there exist z ∈ X and 0 < ε < δ such that
dist(z, ζ) ≤ δ,
sup f ≤ 2f (z),
εf (z) ≥
Bε (z)
δf (ζ)
.
2
Lemma 4.5.4 Let Jν be a sequence of almost complex structures on M converging
to J in the C ∞ -topology and let uν : C∪{∞} → M be a sequence of Jν -holomorphic
curves with bounded energy supν E(uν ) = c < ∞. Then every rigid singular point
z for uν has mass m(z) ≥ ~.
Proof: This is proved by a rescaling argument as in the proof of Theorem 4.3.2.
Assume without loss of generality that 0 is a rigid singular point of uν . Then there
exists a sequence ζν → 0 such that |duν (ζν )| → ∞. Choose a sequence δν > 0 such
that
δν → 0,
δν |duν (ζν )| → ∞.
Now use Lemma 4.5.3 with X = C, f = |duν |, ζ = ζν , and δ = δν to obtain
sequences zν ∈ C and 0 < εν < δν such that zν → 0 and
sup |duν | ≤ 2|duν (zν )|,
εν |duν (zν )| → ∞.
Bεν (zν )
Denote
cν = |duν (zν )|,
Rν = εν cν
and consider the Jν -holomorphic curves vν : BRν → M defined by
vν (z) = uν (zν + cν −1 z).
These satisfy
sup |dvν | ≤ 2,
|dvν (0)| = 1,
Rν → ∞,
E(vν ; BRν ) ≤ c.
BRν
Hence the sequence vν has a subsequence (still denoted by vν ) which converges in
the C ∞ -topology to a J-holomorphic curve v : C → M . The energy of v is finite
55
and hence it follows from the removable singularity theorem that v extends to a
J-holomorphic sphere v : CP 1 = C ∪ {∞} → M . Since |dv(0)| = 1 this sphere must
be nonconstant and hence, by Lemma 4.5.2,
E(v) ≥ ~.
Since v(z) = limν→∞ uν (zν + cν −1 z) we have
E(v; BR ) = lim E(vν ; BR/cν (zν )) ≤ lim inf E(uν ; Bε ).
ν→∞
ν→∞
for every R > 0 (arbitrarily large) and every ε > 0 (arbitrarily small). The last
inequality follows from the fact that R/cν converges to zero and zν converges to 0.
Now take the limit R → ∞ to obtain
~ ≤ lim inf E(uν ; Bε ).
ν→∞
Here we have chosen a subsequence of uν . But the same argument shows that
every subsequence of uν has a further subsequence with this property and hence
mε (0) ≥ ~. Since ε > 0 was chosen arbitrarily this proves the lemma.
2
Lemma 4.5.5 Let Jν be a sequence of almost complex structures on M converging
to J in the C ∞ -topology and let uν : C∪{∞} → M be a sequence of Jν -holomorphic
curves with bounded energy
sup E(uν ) = E < ∞.
ν
Then there exists a subsequence (still denoted by uν ) which has only finitely many
singular points z 1 , . . . , z k and these are all tame with positive mass m(z j ) ≥ ~.
The subsequence can be chosen to converge uniformly with all derivatives on every
compact subset of CP 1 − {z 1 , . . . , z k } to a J-holomorphic curve u : CP 1 → M with
energy
k
X
E(u) = E −
m(z j ).
j=1
Proof: By assumption there exists a constant c > 0 such that E(uν ) ≤ c for every
ν. Hence it follows from Lemma 4.5.4 that the number of rigid singular points is
bounded above by c/~. Now choose a subsequence such that all its singular points
are rigid. Such a subsequence must exist because if there is any nonrigid singular
point left we may choose a further subsequence for which this singular point becomes
rigid. This process must stop after finitely many steps since otherwise there would
be a subsequence with arbitrarily many rigid singular points. By Lemma 4.5.4 this
is impossible.
Now let z 1 , . . . , z k be the singular points of the subsequence uν and assume that
they are all rigid. By definition of singular point the sequence of derivatives duν is
uniformly bounded on every compact subset of CP 1 − {z 1 , . . . , z k }. Passing to a
further subsequence we may assume that uν converges uniformly with all derivatives
on every compact subset of Σ − {z 1 , . . . , z k }. By the removable singularity theorem
the limit extends to a J-holomorphic curve u : CP 1 → M .
56
Now fix a sufficiently small number ε > 0 and choose a further subsequence
such that the limit mε (z j ) = limν→∞ E(uν ; Bε (z j )) exists for every j. Then this
limit exists for every sufficiently small ε > 0 and hence the singular points z j are
all tame for uν . Now for ε > 0 sufficiently small
X
E(u; C − ∪j Bε (z j )) = lim E(uν ; C − ∪j Bε (z j )) = E −
mε (z j )
ν→∞
j
Take the limit ε → 0 to obtain the formula E(u) = E −
P
j
m(z j ).
2
The phenomenon described in the previous lemma already occurs in the context
of rational maps u : CP 1 → CP 1 . For example, the reader might like to consider
a sequence of rational functions of degree 2 with a pole and a zero cancelling each
other out in the limit.
Proof of Theorem 4.4.3: Passing to a suitably reparametrized subsequence (as
in the proof of Theorem 4.3.2) we may assume that uν (1/z) converges uniformly
on compact subsets of C to a nonconstant J-holomorphic curve v 0 : C ∪ {∞} → M .
Hence z = 0 is the only possible singular point of the sequence uν and in view of
Lemma 4.5.5 we may assume it is tame. Denote the mass of this singular point by
Z
m0 = lim mε ,
mε = lim
u∗ν ω.
ν→∞
Bε
ε→0
We will write u for the limit of uν on C − {0} ∪ ∞. Thus, u(z) = v 0 (1/z). Let
us now examine the behaviour of the sequence uν near the singular point z = 0 in
more detail. By Lemma 4.5.2 we have m0 ≥ ~ and hence for every ν there exists a
number δν > 0 such that
Z
~
u∗ν ω = m0 − .
2
Bδν
By definition of the mass m0 the sequence δν converges to 0. Consider the sequence
of J-holomorphic maps vν : C → M defined by
vν (z) = uν (δν z).
We shall prove that there exists a subsequence (still denoted by vν ) such that the
following holds.
(i) The singular set {w1 , . . . , wN } of the subsequence vν is finite and tame and is
contained in the open unit disc B1 ⊂ C.
(ii) The subsequence vν converges with all derivatives uniformly on every compact
subset of C − {w1 , . . . , wN } to a nonconstant J-holomorphic curve v : C → M
with finite energy.
(iii) The energy of v and the masses of the singularities w1 , . . . , wN are related by
E(v) +
N
X
j=1
m(wj ) = m0 .
57
(iv) v(∞) = u(0).
Once this is proved the theorem follows easily by induction. First note that by
definition of m0 and δν
~
lim sup E(vν ; A(1, R)) ≤
2
ν→∞
for every R ≥ 1. Hence there can be no bubbling of holomorphic spheres outside
the unit ball and so statements (i) and (ii) follow from Lemma 4.5.5. Now we shall
prove that the limit curve v : C → M satisfies
E(v; C − B1 ) =
~
.
2
We already have proved that E(v; C − B1 ) ≤ ~/2. To prove the converse choose
a sequence εν > 0 such that E(uν ; Bεν ) = m0 . Then it follows again from the
definition of m0 that εν → 0. Now consider the sequence wν (z) = uν (εν z). It
follows as above that E(wν ; A(1, R)) converges to zero for any R > 1. This implies
that E(wν ; A(δ, 1)) must also converge to zero for any δ > 0 since otherwise a
subsequence of wν would converge to a nonconstant J-holomorphic curve which is
constant for |z| ≥ 1 but such a curve does not exist. Since
E(wν ; A(δν /εν , 1)) = E(uν ; A(δν , εν )) =
~
2
it follows that δν /εν converges to 0. Now, by Lemma 4.5.1, there exists a T0 > 0
such that for T > T0
E(uν ; A(eT δν , e−T εν )) ≤
c~
c
E(uν ; A(δν , εν )) =
.
T
T2
Pick any number α < 1 and choose T so large that 1 − c/T > α. Then the energy
of uν in the union of the annuli A(δν , eT δν ) and A(e−T εν , εν ) must be at least
α~/2. But the energy of uν in A(e−T εν , εν ) converges to 0 while the energy of uν
in A(δν , eT δν ) converges to E(v; A(1, eT )). Hence E(v; A(1, eT )) ≥ α~/2. Since
α < 1 was chosen arbitrarily it follows that E(v, C − B1 ) = ~/2 as claimed.
Statement (iii) now follows from Lemma 4.5.5 for the curves vν : B1 → M with
constant energy EB1 (vν ) = m0 − ~/2. Hence it remains to prove that v(∞) = u(0).
We shall prove that the collar uν (z) for Rδν < |z| < ε converges uniformly as ε → 0,
R → ∞, and ν ≥ ν(ε, R) → ∞. To see this note that
def
E(ε, R) = lim E(uν ; A(Rδν , ε)) = E(u; Bε ) + E(v; C − BR ).
ν→∞
In fact E(uν ; Bε ) converges to E(u; Bε ) + m0 as ν tends to ∞ and, by definition of
δν , it follows that E(uν ; A(δν , ε)) converges to E(u; Bε ) + ~/2. On the other hand,
E(uν ; A(δν , Rδν )) converges to E(v; A(1, R)) = ~/2 − E(v; C − BR ). Subtracting
these two limits gives the required formula above.
Now E(ε, R) converges to zero as ε → 0 and R → ∞. Hence it follows from
Lemma 4.5.1 that for T > 0 and ν > 0 sufficiently large we have
Z 2π
p
dist(uν (Rδν eT +iθ ), uν (εe−T +iθ )) dθ ≤ c E(ε, R).
0
58
Taking the limit ν → ∞ we obtain
Z 2π
p
dist(v(ReT +iθ ), u(εe−T +iθ )) dθ ≤ c E(ε, R).
0
Here the constants T and c are independent of ε and R. Since E(ε, R) → 0 for
ε → 0 and R → ∞ we obtain v(∞) = u(0) as required. Theorem 4.4.3 now follows
easily by induction.
2
Chapter 5
Compactification of Moduli
Spaces
Our goal in this chapter is to explain the compactification C(A, J) of the moduli
space C(A, J) = M(A, J)/G of all J-holomorphic spheres which represent the class
A. Roughly speaking, this compactification is obtained by adding the cusp-curves
to the space C(A, J). In order for this compactification to be useful, it should
carry a fundamental homology class. This will be the case if the set of cusp-curves
which we must add has dimension at least 2 less than that of C. Unfortunately, it
is not known whether this is true for an arbitrary symplectic manifold. However,
we will see that it does hold for manifolds which satisfy a certain positivity (or
monotonicity) condition. This is an important condition, and so we will begin by
explaining it.
Our present approach modifies and streamlines the discussion in McDuff [46]
and Ruan [64]. The main results are the Theorems 5.2.1, 5.3.1 and 5.4.1. Their
proofs are deferred until Chapter 6.
5.1
Semi-positivity
Let (M, ω) be a symplectic manifold and let K > 0. An ω-compatible almost
complex structure J ∈ J (M, ω) is called K-semi-positive if every J-holomorphic
u : CP 1 → M with energy E(u) ≤ K has nonnegative Chern number
Rsphere
∗
u c1 ≥ 0.1 The almost complex structure J is called semi-positive if it is Ksemi-positive for every K. This means that every J-holomorphic curve u : CP 1 →
M has nonnegative Chern
R number. The notions of K-positive and positive are
defined similarly with u∗ c1 > 0. Denote by J+ (M, ω, K) the set of all K-semipositive almost complex structures on M which are ω-compatible and by
\
J+ (M, ω) =
J+ (M, ω, K)
K>0
the set of semi-positive ω-compatible almost complex structures. Note that this
set may be empty. The manifold (M, ω) is called weakly monotone if for every
1 Here
we are identifying the first Chern class c1 with a representing 2-form.
59
60
CHAPTER 5. COMPACTIFICATION OF MODULI SPACES
spherical homology class A ∈ H2 (M, Z)
ω(A) > 0,
c1 (A) ≥ 3 − n
=⇒
c1 (A) ≥ 0.
It is called monotone if there exists a number λ > 0 such that
ω(A) = λc1 (A)
for A ∈ π2 (M ). Note that every monotone symplectic manifold is weakly monotone.
A complex manifold (M, J) (of complex dimension n) is called a Fano variety if its
anti-canonical bundle K ∗ = Λ0,n T ∗ M is ample. This is equivalent to the existence
of a Kähler metric such that the Kähler form ω is monotone.
Remark 5.1.1 In Chapters 5 and 6 we shall work with the space J (M, ω) of
almost complex structures which are ω-compatible. However, all the transversality
and compactness theorems remain valid if we take interpret J (M, ω) as the space of
all ω-tame almost complex structures, replacing the word ω-compatible by ω-tame
wherever it occurs.
Lemma 5.1.2 Let M be a compact 2n-dimensional manifold. Then for all K > 0
the set
(ω, J) ∈ Ω2 (M ) × J (M, ω) | dω = 0, ω n 6= 0, J ∈ J+ (M, ω, K)
is open in the space of all compatible pairs (ω, J) with respect to the C 1 -topology.
In particular, for every symplectic form ω the set J+ (M, ω, K) is open in J (M, ω)
with respect to the C 1 -topology.
Proof: Let ων be a sequence of symplectic forms on M which converge to a
symplectic form ω in the C 1 -topology. Let Jν ∈ J (M, ων ) be a sequence of almost
complex structures on M which are not K-semi-positive with respect to ων . Assume
that Jν converges to J ∈ J (M, ω) in the C 1 -topology. Then there exists a sequence
uν : CP 1 → M of Jν -holomorphic curves with c1 (uν ) < 0 and E(uν ) ≤ K. By
Theorem 4.4.3 uν has a subsequence which converges weakly (in the W 2,p -topology)
to a cusp-curve u = (u1 , . . . , uk ). One of the curves uj must have negative Chern
number and they all have energy E(uj ) ≤ K. Hence J is not K-semi-positive with
respect to ω.
2
In general, the set of semi-positive almost complex structures on a compact
symplectic manifold will not be open. If J is semi-positive then the size of a
neighbourhood which consists of K-semi-positive structures may depend on K.
However, the next lemma shows that if the manifold (M, ω) is weakly monotone
then the set J+ (M, ω) is dense and path-connected.
Lemma 5.1.3 Assume that (M, ω) is a weakly monotone compact symplectic manifold. Then the set J+ (M, ω) contains a path-connected dense subset. The set
J+ (M, ω, K) is open, dense, and path connected for every K.
Proof: We first prove that every almost complex structure J ∈ Jreg (M, ω) which
is regular in the sense of Definition 3.1.1 is semi-positive. To see this let J ∈
Jreg (M, ω) and A ∈ H2 (M, Z) with c1 (A) < 0. We must prove that M(A, J) = ∅.
5.1. SEMI-POSITIVITY
61
Assume otherwise that M(A, J) 6= ∅. Since every curve u ∈ M(A, J) is simple we
must have ω(A) > 0 and, since (M, ω) is weakly monotone this implies
c1 (A) ≤ 2 − n.
But the moduli space M(A, J)/G has dimension
dim M(A, J)/G = 2n + 2c1 (A) − 6
and the above condition on c1 (A) shows that this number is negative. This contradicts our assumption that M(A, J) be nonempty. Thus we have proved that
Jreg (M, ω) ⊂ J+ (M, ω). Now the set Jreg (M, ω) need not be connected. However,
Theorem 3.1.3 shows that any two points J0 , J1 ∈ Jreg (M, ω) can be connected by
a path [0, 1] → J (M, ω) : λ 7→ Jλ such that the quotient space M(A, {Jλ }λ )/G is
a manifold of dimension
dim M(A, {Jλ }λ )/G = 2n + 2c1 (A) − 5.
If c1 (A) < 0 it follows again that this dimension is negative and so for every λ the
space M(A, Jλ ) must be empty. Hence Jλ ∈ J+ (M, ω) for every λ. This proves
the lemma.
2
The condition that (M, ω) be weakly monotone depends only on the homotopy
classes of ω and c1 (M ), and is the relevant positivity condition to consider in
the case of a symplectic manifold. If one starts with a Kähler manifold though,
one is often more interested in the properties of the underlying complex manifold
(M, J0 ) than in the cohomology class of the chosen Kähler form ω. We will see in
Proposition 7.2.3 below that our results apply to complex manifolds (M, J0 ) such
that J0 is compatible with some weakly monotone Kähler (or symplectic) form ω.
Remark 5.1.4 It is not hard to check that a symplectic manifold (M, ω) is weakly
monotone if and only if one of the following conditions is satisfied.
• (M, ω) is monotone.
• either c1 (A) = 0 for every A ∈ π2 (M ), or ω(A) = 0 for every A ∈ π2 (M ).
• The minimal Chern number N , defined by hc1 , π2 (M )i = N Z where N ≥ 0,
is at least N ≥ n − 2.
These conditions are not mutually exclusive. The last condition shows that every
symplectic manifold of dimension less than or equal to 6 is weakly monotone. The
case when ω(A) is always zero is not very interesting in the present context, since
obviously there cannot be any J-holomorphic spheres in such manifolds. However,
there could be J-holomorphic curves of higher genus.
2
As an example, let M bePa product of a number of projective spaces, M =
CP n1 ×· · ·×CP nk , with form i ai τi , where τi is the usual Kähler form, normalised
so that it integrates to π over each complex line. Then M is monotone only if all
the ai are equal. However, it is easy to see that the standard complex structure J0
is semi-positive. Hence, for each K > 0 there exists a neighbourhood of J0 which
consists entirely of K-semi-positive almost complex structures.
62
5.2
The image of the evaluation map
We now describe a first result concerning the compactification of the moduli space.
There are different ways in which this result can be stated. Instead of specifying a
topology on the closure C of the space of unparametrized curves, we have chosen to
express the fact that C is closed by stating that it has closed image in M , since this
is what is needed for the applications we have in mind. Thus we consider the image
of our spaces of curves by appropriate evaluation maps. We will use the letter W
to describe those moduli spaces which are the natural domains of definition for
evaluation maps.
Recall from Chapter 3 that the reparametrization group G = PSL(2, C) acts on
M(A, J) × CP 1 by
φ · (u, z) = (u ◦ φ−1 , φ(z)).
The quotient space
W(A, J) = M(A, J) ×G CP 1
is the domain of the evaluation map
e : W(A, J) → M,
e(u, z) = u(z).
We shall denote the image of this map by
[
X(A, J) = e(W(A, J)) =
u(CP 1 ).
u∈M(A,J)
Our goal is to describe the closure X(A, J) as a kind of stratified space with open
stratum X(A, J) and all other strata of codimension at least 2. We must prove that
the lower dimensional cover all possible limit points of sequences e(uν , zν ) where
the sequence [uν , zν ] does not converge in W(A, J). These other strata are defined,
roughly speaking, as the images of evaluation maps on spaces of simple cusp-curves
which represent the class A. A cusp-curve
C = C1 ∪ · · · ∪ CN
is called simple2 if all its components C i are distinct and none are multiply-covered.
More precisely, a simple cusp-curve is a collection of simple J-holomorphic spheres
ui : CP 1 → M
such that their images form a connected set and ui 6= uj ◦ φ for i 6= j and any
fractional linear transformation φ ∈ G. Observe that every cusp-curve can be
simplified to a simple curve, by getting rid of repeated components and replacing
each multiply covered component by its underlying simple component. Clearly, this
process does not change the set of points which lie on the curve, although it will
change its homology class.
2 In
[46] and [64] such curves were called reduced.
5.2. THE IMAGE OF THE EVALUATION MAP
63
Theorem 5.2.1 Let (M, ω) be a compact symplectic manifold.
(i) For every ω-compatible almost complex structure J ∈ J (M, ω) there exists a
finite collection of evaluation maps eD : W(D, J) → M such that
[
\
e(W(A, J) − K) ⊂
eD (W(D, J)).
K⊂W(A,J)
K compact
D
(ii) There exists a set Jreg = Jreg (M, ω, A, 1) ⊂ J (M, ω) of the second category
such that the set W(D, J) is a smooth oriented σ-compact manifold of dimension
dim W(D, J) = 2n + 2c1 (D) − 2N − 2
for J ∈ Jreg . Here N is the number of components of the class D.
(iii) Assume that A is not a multiple class mB where m > 1 and c1 (B) = 0. If
J ∈ J+ (M, ω, K) ∩ Jreg for some K > ω(A) then
dim W(D, J) ≤ dim W(A, J) − 2
for every D 6= A.
In the terminology of Section 7.1, the above theorem asserts that the evaluation
map e = eJ : W(A, J) → M determines a pseudo-cycle for a generic almost complex
structure J. This means that its image can be compactified by adding pieces of
codimension at least 2, and hence that it carries a fundamental class. We shall see
below that this homology class is independent of the almost complex structure J
used to define it. (See Proposition 7.2.2 with p = 1.) The proof involves choosing
a regular path of almost complex structures Jλ running from J0 to J1 such that
the corresponding space W(D, {Jλ }λ ) is a smooth manifold. This manifold forms a
cobordism from W(D, J0 ) to W(D, J1 ) and there is an analogue of Theorem 5.2.1
for the evaluation map on W(D, {Jλ }λ ). In short, the evaluation maps eJ0 and eJ1
are bordant as pseudo-cycles.
The framing D
We now give the relevant definitions for the notation in Theorem 5.2.1. The letter
D denotes an effective framed class. This means that
D = {A1 , . . . , AN , j2 , . . . , jN }
where Ai ∈ H2 (M, Z) and jν is an integer with 1 ≤ jν ≤ ν−1. The homology classes
Ai are effective in the sense that they each have J-holomorphic representatives and
can be the classes of the components of a simplified cusp-curve C which represents
the class A. For example there is a necessary condition ω(Ai ) ≤ ω(A) for all i and
hence it follows from Corollary 4.4.4 that there is only a finite number of possible
framed classes D. The “framing” of D describes how the different components
intersect. The number jν indicates that the cusp-curve u = (u1 , . . . , uN ) must
satisfy
uν (CP 1 ) ∩ ujν (CP 1 ) 6= ∅.
64
Note that the class D of a cusp-curve can never equal {A} itself since either N > 1
or D = {A1 } where A = mA1 for some m > 1.
We will see in Chapter 6 that, in order to make W(D, J) into a manifold,
we must restrict to considering simple cusp-curves. In fact, we saw already in
Chapter 3 why we must avoid multiply covered
curves, and repeated components
P
can also cause trouble. This means that i Ai need not equal A, since the process
of simplification changes the homology class. However, there are integers mi ≥ 1
such that
X
mi Ai = A.
i
If J is K-semi-positive then c1 (Ai ) ≥ 0 for all i and so
X
c1 (D) =
c1 (Ai ) ≤ c1 (A).
i
This is the basic reason why statement (iii) in Theorem 5.2.1 holds in this case.
Observe also that the set of points which lie on C is unchanged if we replace C by
its simple reduction. Hence restricting to simple C does not affect the validity of
(i).
We remark that the framing (or intersection pattern) associated to D describes
enough of the intersection pattern of the different components of C to ensure that
C is connected. For example, one might insist that the A2 -curve intersects the A1 curve, and that the A3 curve also intersects the A1 -curve, and so on. Because we
do not insist on describing the full intersection pattern, our approach is somewhat
simpler than that in [64]. Also, there are cusp-curves which belong to more than
one set C(D, J). Further details may be found in Chapter 6. Elements of C(D, J)
are called simple cusp-curves of type D.
The spaces W(D, J)
According to our conventions, the spaces W(D, J) are the correct spaces on which
evaluation maps are defined. A precise definition will be given in Chapter 6. As an
example consider the case D = {A, B} with A 6= B. In this case there is only one
possible intersection pattern, namely the A-curve must intersect the B-curve. Fix
a point z0 ∈ CP 1 and define G0 = {φ ∈ G | φ(z0 ) = z0 }. The group G0 × G0 acts
on the space
M(D, J) = {(u, v) | u ∈ M(A, J), v ∈ M(B, J), u(z0 ) = v(z0 )} .
This space is the inverse image of the diagonal ∆ ⊂ M × M under the map
M(A, J) × M(B, J) : (u, v) 7→ (u(z0 ), v(z0 )) and hence, by a general position
argument with generic J, will have dimension
dim M(D, J) = 2n + 2c1 (A).
Now the group G0 × G0 acts on the space M(D, J) × CP 1 in two ways, namely
−1
by (φ1 , φ2 ) · (u, v, z) = (u ◦ φ−1
1 , v ◦ φ2 , φj (z)) for j = 1, 2. This leads to two
different quotient spaces W1 (D, J) and W2 (D, J) with corresponding evaluation
maps eA , eB : Wj (D, J) → M defined by
eA (u, v, z) = u(z),
eB (u, v, z) = v(z).
5.3. THE IMAGE OF THE P-FOLD EVALUATION MAP
65
We define W(D, J) = W1 (D, J) ∪ W2 (D, J). Since the group G0 × G0 is 8dimensional and in both cases acts freely on M(D, J) × CP 1 we obtain
dim W(D, J) = 2n + 2c1 (A) − 6.
This is precisely 2 less than the dimension of W(A, J), thus confirming the dimension formula in statement (ii) of Theorem 5.2.1.
Granted these definitions, part (i) of Theorem 5.2.1 follows immediately from
Gromov’s compactness theorem. The main additional technical point, which one
needs in order to control the dimension of the spaces W(D, J), is a proof that for
generic J the evaluation maps
eA : M(A, J) × S 2 → M,
eB : M(B, J) × S 2 → M
are transverse. It is here that the condition u 6= v ◦ φ becomes important. The
details will be carried out in Chapter 6.
5.3
The image of the p-fold evaluation map
In order to define the Gromov-Witten invariants we must also compactify the image
X(A, J, p) of the p-fold evaluation map
ep : W(A, J, p) → M p
defined on the quotient space W(A, J, p) = M(A, J) ×G (CP 1 )p . Here the group
G = PSL(2, C) acts on M(A, J) × (CP 1 )p by
φ · (u, z1 , . . . , zp ) = (u ◦ φ−1 , φ(z1 ), . . . , φ(zp ))
and the evaluation map is given by
ep (u, z1 , . . . , zp ) = (u(z1 ), . . . , u(zp )).
Denote the image of this evaluation map by X(A, J, p) = ep (W(A, J, p)) ⊂ M p .
Essentially the same result applies in this case. Suppose that the A-curves uν :
CP 1 → M converge to a simple cusp-curve C of type
D = {A1 , . . . , AN , j2 , . . . , jN }
and consider possible limit points of the p-tuples (uν (z1ν ), . . . , uν (zpν )). After passing to a subsequence, if necessary, the the points uν (zjν ) will converge to a point
zj on some component of C. Hence we must consider all possible configurations
of limit points zj ∈ CT (j) . Thus the domain of our p-fold evaluation maps will be
determined by pairs (D, T ) where D is a framed class of simple reduced curves as
before and the label T is a map from {1, . . . , p} → {1, . . . , N }. For each such pair
(D, T ) there will be an evaluation map
eD,T : W(D, T, J, p) → M p
whose image X(D, T, J, p) = eD,T (W(D, T, J, p)) ⊂ M p consists of all p-tuples of
points (x1 , . . . , xp ) such that xj ∈ CT (j) where C = ∪i Ci runs through all simple
cusp-curves of type D. These images will form the closure of the set X(A, J, p).
66
Theorem 5.3.1 Let (M, ω) be a compact symplectic manifold.
(i) For every J ∈ J (M, ω) there exists a finite collection of evaluation maps eD,T :
W(D, T, J, p) → M p such that
[
\
ep (W(A, J, p) − K) ⊂
eD,T (W(D, T, J, p)).
K⊂W(A,J,p)
K compact
D,T
(ii) There exists a set Jreg = Jreg (M, ω, A, p) ⊂ J (M, ω) of the second category
such that the set W(D, T, J, p) is a smooth oriented σ-compact manifold of
dimension
dim W(D, T, J, p) = 2n + 2c1 (D) + 2p − 2N − 4
for J ∈ Jreg . Here N is the number of components of the class D.
(iii) Assume that A is not a multiple class mB where m > 1 and c1 (B) = 0. If
J ∈ J+ (M, ω, K) ∩ Jreg for some K > ω(A) then
dim W(D, T, J, p) ≤ dim W(A, J, p) − 2
for all D 6= A.
This may be proved by the same methods as the previous theorem. The main
problem is in correctly defining the sets W(D, T, J, p) and this will be done in the
Section 6.5. Again, this theorem asserts that for a generic almost complex structure
J the evaluation map ep : W(A, J, p) → M p determines a pseudo-cycle in the sense
of Section 7.1. Hence its image carries a fundamental homology class, and we shall
see that this class is independent of the choice of J (see Proposition 7.2.2). This
suffices to define the Gromov invariant Φ.
5.4
The evaluation map for marked curves
There is an alternative way of defining a p-fold evaluation map, namely by fixing a
p-tuple
z = (z1 , . . . , zp ) ∈ (CP 1 )p
of distinct points in CP 1 and evaluating a (parametrized) J-holomorphic curve u
at the points zj . Hence define the evaluation map
ez = eA,J,z : M(A, J) → M p
by
ez (u) = (u(z1 ), . . . , u(zp )).
This map is used to define the Gromov-Witten invariant Ψ (which Ruan called
Φ̃) and will play an important role in the definition of quantum cohomology. The
natural assumption for the definition of this invariant is p ≥ 3 since three points
on a nonconstant curve determine its parametrization. In fact in the case p = 3 we
shall see that both invariants agree and no new construction is required. However,
5.4. THE EVALUATION MAP FOR MARKED CURVES
67
for p > 3 there are genuine differences. This will already be apparent in the proof
of the following compactness theorem for the images
Y (A, J, z) = ez (M(A, J)) ⊂ M p .
It is worth pointing out that for any triple z = (z1 , z2 , z3 ) of distinct points in
CP 1 we have Y (A, J, z) ⊂ X(A, J, 3) and the closures of both sets agree. The sets
themselves need not be the same since in the definition of X the points on which u
is evaluated are not required to be distinct.
For p > 3 there is still an inclusion Y (A, J, z) ⊂ X(A, J, p). However the set
Y (A, J, z) will have strictly lower dimension. To get a picture of how these spaces
fit together for different choices of z, consider the diagram
M(A, J) × (CP 1 )p
π↓
(CP 1 )p
ẽp
→
Mp
The set Y (A, J, z) is simply the image of the fiber π −1 (z) under the obvious evaluation map ẽp . Thus the sets Y (A, J, z) may be thought of as forming the fibers of
a (singular) fibration of X(A, J, p) of codimension 2p − 6.
As before our goal is to prove that the boundary of the set Y (A, J, z) is a finite
union of lower dimensional strata. Thus we must investigate how the images of the
manifolds W(D, T, J, p) which compactify X(A, J, p) intersect the fibers Y (A, J, z).
In particular, in order to give an accurate description of the possible limiting positions of the images of the points zi , we must keep track of the points at which the
bubbles occur. For example, if uν is a sequence in M(A, J) which develops a bubble
at z1 , then the sequence of points uν (z1 ) may have subsequences which converge
to any point of the bubble at z1 . It follows that the whole bubble is contained
in Y (A, J, z). Multiply-covered components of the limit also pose new problems.
The upshot is that we can only prove that a good compactification exists under
additional assumptions.
More precisely, we will assume that our almost complex structure J ∈ J (M, ω)
and the homology class A satisfy the following condition.
(JAp ) Every J-effective homology class B ∈ H2 (M, Z) has Chern number
c1 (B) ≥ 2.
Moreover, if A = mB ∈ H2 (M, Z) is the m-fold multiple of a J-effective
homology class B ∈ H2 (M, Z) with m > 1 then either c1 (B) ≥ 3 or p ≤ 2m.
Here a homology class A ∈ H2 (M, Z) is called J-effective if it can be represented
by a J-holomorphic sphere u : CP 1 → M . In particular a J-effective homology
class is necessarily spherical. These assumptions are stronger than what is actually
needed. For example in the case p = 3 it suffices to assume
(JA3 ) Every J-effective homology class B ∈ H2 (M, Z) has Chern number
c1 (B) ≥ 0.
homology class B ∈ H2 (M, Z) with m > 1 then c1 (B) ≥ 1.
68
This is precisely the condition of Theorem 5.3.1 (iii). In the case p = 4 only the
following condition is required.
(JA4 ) Every J-effective homology class B ∈ H2 (M, Z) has Chern number
c1 (B) ≥ 1.
homology class B ∈ H2 (M, Z) with m > 1 then c1 (B) ≥ 2.
For general p the above condition can also be weakened but the precise condition
required with the methods of this book is somewhat complicated to state. So we
content ourselves with the above formulation.
We point out that for the definition of quantum cohomology it suffices to consider the cases p = 3 and p = 4. So in this case we may simply assume (JA3 )
and (JA4 ) for all classes A. These conditions will be satisfied, for example, if
(M, ω) is monotone with minimal Chern number at least 2.
Now it would be much nicer, of course, if statement (iii) of the following theorem
could be proved for generic J without any further assumptions on J. But this would
require a better understanding of the behaviour of multiply covered J-holomorphic
curves with Chern number less than or equal to 2 and with the present techniques
it is not clear how to do this.
Theorem 5.4.1 Let (M, ω) be a compact symplectic manifold and fix a homology
class A ∈ H2 (M ) and a p-tuple z = (z1 , . . . , zp ) ∈ (CP 1 )p of distinct points in CP 1 .
(i) For every J ∈ J (M, ω) there exists a finite collection of evaluation maps
eD,T,z : V(D, T, J, z) → M p
such that
\
ez (M(A, J) − K) ⊂
K⊂M(A,J)
K compact
[
eD,T,z (V(D, T, J, z)).
D,T
(ii) There exists a set Jreg = Jreg (M, ω, A, z) ⊂ J (M, ω) of the second category such that the set V(D, T, J, z) is a finite dimensional smooth oriented
σ-compact manifold for every J ∈ Jreg .
(iii) Assume J ∈ Jreg and the pair (J, A) satisfies (JAp ). Then
dim V(D, T, J, z) ≤ dim M(A, J) − 2
for all D 6= A.
This theorem is proved in Section 6.6. As before, this theorem asserts that
for a generic almost complex structure J the evaluation map ez : M(A, J) → M p
determines a pseudo-cycle in the sense of Section 7.1 and so its image carries a
fundamental homology class. We shall see below that this class is independent
of the point z ∈ (CP 1 )p and of the almost complex structure J used to define
it (Lemma 7.4.1). The proof of this fact again relies on the construction of a
5.4. THE EVALUATION MAP FOR MARKED CURVES
69
bordism from ez0 : M(A, J0 ) → M p to ez1 : M(A, J1 ) → M p . This requires an
analogue of Theorem 5.4.1 for paths along the following lines. Given a path [0, 1] →
(CP 1 )p : λ 7→ zλ and regular almost complex structures J0 ∈ Jreg (M, ω, A, z0 ) and
J1 ∈ Jreg (M, ω, A, z1 ) in the sense of Theorem 5.4.1, we must find a path of almost
complex structures Jλ ∈ J (M, ω) such that the moduli space
M(A, {Jλ }λ ) = {(λ, u) | u ∈ M(A, Jλ )}
is a smooth manifold. Now consider the closure of the image of the evaluation map
M(A, {Jλ }λ ) → M p : (λ, u) 7→ ezλ (u).
For a generic path Jλ this closure will again be obtained by adding pieces of codimension at least 2. The details are the same as in the proof of Theorem 5.4.1 and
this gives rise to the required bordism between the pseudo-cycles ez0 : M(A, J0 ) →
M p and ez1 : M(A, J1 ) → M p . This suffices to construct the Gromov-Witten
invariant Ψ.
Remark 5.4.2 (i) The strong condition (JAp ) is used only to deal with multiplycovered components of the limiting cusp-curve. If there is some way to ensure
that such multiply-covered curves cannot appear in the limit, then it suffices to assume that the manifold is weakly monotone. We shall exploit this
observation in Section 9.3.
(ii) We emphasize that the notion of a regular almost complex structure in Theorem 5.4.1 depends on the point z which we have fixed to begin with. In
practice it may be useful to fix instead an almost complex structure J which
is regular in the sense of Theorem 5.3.1, and then vary the point z to achieve
transversality.
2
70
Chapter 6
Evaluation Maps and
Transversality
This chapter contains the proofs of the main results of Chapter 5 and the reader
may wish to go directly to Chapter 7. As we explained in Chapter 5 the pieces
which must be added to compactify M(A, J) correspond to the cusp-curves. Our
first task is to show that these cusp-curves appear in finite-dimensional families,
and to calculate their dimension. This is accomplished in the first four sections.
The last two sections contain the proofs of the compactness theorems of Chapter 5.
6.1
Evaluation maps are submersions
Every embedded symplectic 2-sphere is a J-holomorphic curve for some almost
complex structure J which is compatible with ω. Since the symplectic condition
is open it follows that the set of all points which lie on embedded J-holomorphic
curves is open. This suggests that the evaluation map e0 : M(A, J ) → M at the
point z0 ∈ CP 1 should be a submersion. Our goal in this section is to prove this
for all simple curves, not just the embedded ones. Here we denote by J = J (M, ω)
the space of almost complex structures on M which are compatible with ω. We
could equally well consider the space of all ω-tame J: the only difference would be
in the formula for the tangent space TJ J . The evaluation map
e0 : M(A, J ) → M
at z0 ∈ CP 1 is defined by e0 (u, J) = u(z0 ).
Theorem 6.1.1 For every point z0 ∈ CP 1 the map e0 : M(A, J ) → M is a
submersion.
Recall from Section 3.4 that the tangent space of M(A, J ) at a pair (u, J) with
∂¯J (u) = 0 is the set of all pairs
(ξ, Y ) ∈ C ∞ (u∗ T M ) × C ∞ (End(T M, J, ω))
71
72
CHAPTER 6. EVALUATION MAPS AND TRANSVERSALITY
which satisfy
Du ξ + 21 Y (u)du ◦ i = 0.
The differential of the evaluation map is obviously given by
de0 (u, J)(ξ, Y ) = ξ(z0 ).
We shall prove that, given any vector v ∈ Tu(z0 ) M there exists a pair (ξ, Y ) ∈
T(u,J) M(A, J ) such that ξ(z0 ) = v. Moreover, we shall prove that ξ can be chosen
with support in an arbitrarily small neighbourhood of z0 . In fact we shall first
construct a local solution ξ of Du ξ = 0 near the point z0 and then modify ξ by a
cutoff function to make it vanish outside a small neighbourhood of z0 . We then have
to compensate for the effect of the cutoff function by introducing an infinitesimal
almost complex structure Y which is supported in a neighbourhood of the image
under u of a small annulus centered at z0 .
Lemma 6.1.2 Given J ∈ J (M, ω) and a J-holomorphic curve u : CP 1 → M there
exists a constant δ > 0 such that for every v ∈ Tu(z0 ) M and every pair 0 < ρ < r < δ
there exists a smooth vector field ξ(z) ∈ Tu(z) M along u and an infinitesimal almost
complex structure Y ∈ C ∞ (End(T M, J, ω)) such that the following holds.
(i) Du ξ + 21 Y (u)du ◦ i = 0.
(ii) ξ(z0 ) = v.
(iii) ξ is supported in Br (z0 ) and Y is supported in an arbitrarily small neighbourhood of u(Br (z0 ) − Bρ (z0 )).
Proof: Recall from the proof of Proposition 3.4.1 that
Du ξ + 21 Y (u)du ◦ i = η ds − J(u)η dt
where
η = ∂s ξ + J(u)∂t ξ + (∂ξ J(u))∂t u + Y (u)∂t u.
Here we denote by z = s + it conformal coordinates on CP 1 and think of ξ(s, t) ∈
R2n as the local coordinate representation of a tangent vector in T M with respect
to a Darboux coordinate chart. Now choose a unitary trivialization Φ(s, t) : R2n →
Tu(s,t) M such that
J(u(s, t))Φ(s, t) = Φ(s, t)J0 ,
where
J0 =
ω(Φv, Φw) = (J0 v)T w
0 −1l
1l
0
,
and T denotes the transpose, and write
ξ(s, t) = Φ(s, t)ξ0 (s, t),
v = Φ(0, 0)v0 ,
∂t u(s, t) = −Φ(s, t)ζ0 (s, t).
Then we have
Du ξ + 21 Y (u)du ◦ i = Φ(η0 ds − J(u)η0 dt)
6.1. EVALUATION MAPS ARE SUBMERSIONS
73
where
η0 = ∂s ξ0 + J0 ∂t ξ0 + A0 ξ0 − Y0 ζ0
and
Y0 = Φ−1 Y (u)Φ,
A0 = Φ−1 (∂s Φ + J(u)∂t Φ + (∂Φ J)(u)) .
The compatibility condition Y (x) ∈ End(Tx M, Jx , ωx ) for x ∈ M translates into
Y0 = Y0T = J0 Y0 J0 .
We must find a matrix-valued function Y0 : B1 → R2n×2n and a vector field ξ0 :
B1 → R2n such that
∂s ξ0 + J0 ∂t ξ0 + A0 ξ0 = Y0 ζ0 ,
ξ0 (0) = v0 .
First note that the condition ∂s ξ0 + J0 ∂t ξ0 = 0 means that ξ0 is holomorphic and
there exists a holomorphic curve through every point v0 (for example the constant
function). Now a standard perturbation argument in Fredholm theory shows that
if A0 is sufficiently small then the equation
∂s ξ0 + J0 ∂t ξ0 + A0 ξ0 = 0
with ξ0 (0) = v0 has a solution ξ0 : B1 → R2n in the unit ball for every value of v0 .
For example this can be proved by considering a regular boundary value problem.
By a rescaling argument this can be extended to arbitrary A0 but with the domain
of the solution ξ0 restricted to Bδ where δ depends on A0 . Now for 0 < ρ < r < δ
choose a cutoff function
β : C → [0, 1]
such that β(z) = 1 for |z| < ρ and β(z) = 0 for |z| > r. Then the function βξ0
satisfies
def
∂s (βξ0 ) + J0 ∂t (βξ0 ) + A0 (βξ0 ) = (∂s β)ξ0 + (∂t β)J0 ξ0 = η0 .
So we must find Y0 such that
Y0 ζ0 = η0
−1
where ζ0 = −Φ ∂t u. Such Y0 exists because each space End(R2n , J0 , ω0 ) acts
transitively on the vectors in R2n . An explicit formula is given by
Y0
=
1
1
η0 ζ0T + ζ0 η0T +
J0 η0 ζ0T + ζ0 η0T J0
|ζ0 |2
|ζ0 |2
hη0 , ζ0 i
−
ζ0 ζ0T + J0 ζ0 ζ0T J0
4
|ζ0 |
hη0 , J0 ζ0 i
−
J0 ζ0 ζ0T − ζ0 ζ0T J0 .
4
|ζ0 |
This function is well defined wherever du(s, t) 6= 0 and satisfies Y0T = Y0 = J0 Y0 J0
and Y0 ζ0 = η0 . Now it follows from Lemma 2.2.3 that the restriction of u to an
(arbitrarily small) annulus Br − Bρ is an embedding.1 For such a choice of ρ and
r there exists a Y ∈ C ∞ (End(T M, J, ω)) such that
Φ(s, t)Y0 (s, t) = Y (u(s, t))Φ(s, t).
1 This
is where we use the hypothesis that u is simple.
74
This function Y has the required properties and is supported in an arbitrarily small
neighbourhood of u(Br − Bδ ). This proves the lemma.
2
Theorem 6.1.1 follows immediately from Lemma 6.1.2. An alternative geometric
proof can be given in terms of the local deformation theory for J-holomorphic curves
as in Nijenhuis and Woolf [57] or McDuff [44], [48], [50]: see [42, Proposition 4.1].
In fact our proof of Lemma 6.1.2 can be viewed as a linearized version of these
deformation results.
6.2
Moduli spaces of N -tuples of curves
Consider the universal moduli space
M(A1 , . . . , AN , J )
of N -tuples of distinct curves. This consists of all (N + 1)-tuples (u1 , . . . , uN , J)
such that J ∈ J , uj ∈ M(Aj , J) and
ui 6= uj ◦ φ for φ ∈ G and i 6= j.
In the case Ai 6= Aj the condition ui 6= uj ◦ φ is automatically satisfied and the
set M(A1 , . . . , AN , J ) is simply the union of the sets of all N -tuples (u1 , . . . , uN )
with uj ∈ M(Aj , J) over all J ∈ J . In the case Ai = Aj for some pair i 6= j we
have deleted the “diagonal” of all N -tuples with ui = uj ◦ φ from the moduli space.
Note that if ui = uj ◦ φ for some i 6= j the corresponding cusp-curve {C 1 , . . . , C N }
(where C i = ui (CP 1 )) is not simple and hence will not appear in the proof of
Theorem 5.2.1 or 5.3.1.
Remark 6.2.1 It follows from Proposition 2.3.2 that the reparametrization group
G × · · · × G = GN acts freely on the space M(A1 , . . . , AN , J ) by
−1
N
φ · (u1 , . . . , uN , J) = (u1 ◦ φ−1
1 , . . . , u ◦ φN , J)
for φ = (φ1 , . . . , φN ) ∈ GN .
2
Following the line of argument in Chapter 3 we introduce the space J ` of almost
complex structures of class C ` where ` ≥ 1. We also choose p > 2 and 1 ≤ k ≤ ` + 1
and denote by
M` (A1 , . . . , AN , J ) = M(A1 , . . . , AN , J ` )
the corresponding universal moduli space where J ∈ J ` and the J-holomorphic
curves uj are of class W k,p . Recall from Chapter 3 that this space is independent
of the choice of k and p.
Proposition 6.2.2 The space M` (A1 , . . . , AN , J ) is a Banach manifold.
Proof: The proof is almost word by word the same as that of Proposition 3.4.1
and we shall content ourselves with summarizing the main points. Denote by X
the space of all smooth maps
u = (u1 , . . . , uN ) : CP 1 → M N
6.3. MODULI SPACES OF CUSP-CURVES
75
for which there exists an N -tuple (z1 , . . . , zN ) ∈ (CP 1 )N which satisfies the conditions of Proposition 2.3.2. This set is open (with respect to the C 1 -topology) in the
space of all smooth maps from CP 1 to M N and it follows from Proposition 2.3.2
that M(A1 , . . . , AN , J ) ⊂ X . In fact, our universal moduli space is the set of zeros
of a section of the bundle E → X with fibers
Eu = Ω0,1 ((u1 )∗ T M ) ⊕ · · · ⊕ Ω0,1 ((uN )∗ T M )
This section F : X → E is given by
F(u1 , . . . , uN , J) = (∂¯J (u1 ), . . . , ∂¯j (uN )).
Now the differential of this section at a zero (u, J) is the operator
DF(u, J) : Tu X ⊕ TJ J → Eu
given by
DF(u, J)(ξ, Y ) = (Du1 ξ1 + 21 Y ◦ du1 ◦ i, . . . , DuN ξN + 21 Y ◦ duN ◦ i).
where ξ = (ξ1 , . . . , ξN ) ∈ Tu X = C ∞ ((u1 )∗ T M ) ⊕ · · · ⊕ C ∞ ((uN )∗ T M ). The proof
that this operator is onto (when considered on the usual Sobolev spaces) is word
by word the same as that of Proposition 3.4.1. The only point of difference is that
the existence of an N -tuple (z1 , . . . , zN ) ∈ (CP 1 )N which satisfies the conditions of
Proposition 2.3.2 is used to prove that the operator DF(u, J) has a dense range.
Further details are left to the reader.
2
Observe that if Ai = Aj for some i 6= j then we can add a point (u, J) =
(u , . . . , uN , J) with ui = uj ◦φ to the moduli space without destroying the manifold
structure provided that the operator DF(u, J) is onto. For example, this will be
the case if the operators Duj are onto for all j. However, adding such curves would
destroy the free action of the group GN on M(A1 , . . . , AN , J ).
1
6.3
Moduli spaces of cusp-curves
Recall that a cusp-curve C which represents the class A is a connected union
C = C1 ∪ · · · ∪ CN
of J-holomorphic curves C j = uj (CP 1 ), which are called its components, such that
the corresponding homology classes Aj = [C j ] have the sum A1 + · · · + AN = A.
Since C is reducible, either N > 1 or C 1 is multiply covered. C is said to be simple
if all its components are distinct and none are multiply-covered. Any curve C can
be simplified by deleting all but one copy of repeated components, and by replacing
any multiply-covered component by its underlying simple curve. Note that both
these operations change the homology class [C]. We will see that to compactify the
set C(A, J) of all simple A-curves we will need to add to it all simplified cusp-curves
which represent A. The following lemma was already used in Remark 4.4.2. The
proof is an elementary exercise.
76
Lemma 6.3.1 The components of C can be ordered so that C 1 ∪ · · · ∪ C k is connected for all k ≤ N .
Each simple cusp-curve C = C 1 ∪ · · · ∪ C N has a type D. This is the set
{A , . . . , AN } of homology classes Aj = [C j ], together with a framing which specifies that certain components of C intersect. We do not need to describe the full
intersection pattern as was done in [64], but just enough of it to ensure that C is connected. Now suppose that the components of of C are ordered as in Lemma 6.3.1.
Then each component C ν must intersect some component C jν where 1 ≤ jν ≤ ν −1,
and so we take the set of integers {j2 , . . . , jN } to be the framing data. For every
framed class
D = {A1 , . . . , AN , j2 , . . . , jN }
1
and each J ∈ J (M, ω) denote by
M(D, J) ⊂ M(A1 , . . . , AN , J) × (CP 1 )2N −2
the moduli space of all points (u, w, z) with w = (w2 , . . . , wN ) ∈ (CP 1 )N −1 , z =
(z2 , . . . , zN ) ∈ (CP 1 )N −1 , and
uj ∈ M(Aj , J)
such that u = (u1 , . . . , uN ) is a simple J-holomorphic cusp-curve with
ujν (wν ) = uν (zν )
for ν = 2, . . . , N . Our first goal in this section is to prove that for a generic
choice of the almost complex structure the moduli space M(D, J) is a smooth
finite dimensional manifold.
Theorem 6.3.2 There exists a set Jreg = Jreg (D) ⊂ J of the second category
such that M(D, J) is a manifold of dimension
dim M(D, J) = 2
N
X
c1 (Aj ) + 2n + 4(N − 1).
j=1
for every J ∈ Jreg .
As before the letter J denotes the space of ω-compatible almost complex structures.
To prove this theorem consider the evaluation map
eD : M(A1 , . . . , AN , J) × (CP 1 )2N −2 → M 2N −2
defined by
eD (u, w, z) = (uj2 (w2 ), u2 (z2 ), . . . , ujN (wN ), uN (zN ))
for u ∈ M(A1 , . . . , AN , J ), w = (w2 , . . . , wN ) ∈ (CP 1 )N −1 , and z = (z2 , . . . , zN )
∈ (CP 1 )N −1 . With this notation the moduli space M(D, J) is the inverse image
of the multi-diagonal
∆N = (x, y) = (x2 , y2 , . . . , xN , yN ) ∈ M 2N −2 | xν = yν
6.3. MODULI SPACES OF CUSP-CURVES
77
under the evaluation map eD . Thus, Theorem 6.3.2 asserts that the set Jreg (D)
of J for which the evaluation map eD is transverse to ∆N has second category in
J (M, ω). To prove this we consider the universal moduli space
M(D, J )
of all points (u, J, w, z) with J ∈ J (M, ω) and (u, w, z) ∈ M(D, J). This space is
defined to be the inverse image of the diagonal ∆N under the extended evaluation
map
eD : M(A1 , . . . , AN , J ) × (CP 1 )2N −2 → M 2N −2
which is defined by the same formula as above. For the case N = 2 the following
crucial transversality result was proved by McDuff in [42].
Proposition 6.3.3 For each framed class D = {A1 , . . . , AN , j2 , . . . , jN } the evaluation map
eD : M(A1 , . . . , AN , J ) × (CP 1 )2N −2 → M 2N −2
is transversal to the multi-diagonal ∆N ⊂ M 2N −2 . Hence
M(D, J ) = e−1
D (∆N ).
is an infinite dimensional Fréchet manifold.
Proof:
map
Fix a (2N − 2)-tuple (w, z) ∈ (CP 1 )2N −2 . It suffices to prove that the
M(A1 , . . . , AN , J ) → M 2N −2 : (u, J) 7→ eD (u, J, w, z)
is transverse to ∆N . The tangent space to M(A1 , . . . , AN , J ) at the point (u, J) =
(u1 , . . . , uN , J) consists of all (ξ, Y ) = (ξ1 , . . . , ξN , Y ) where
ξν ∈ C ∞ ((uν )∗ T M ),
Y ∈ C ∞ (End(T M, J, ω)),
and
Duν ξν + 21 Y (uν )duν ◦ i = 0.
The differential of the map (u, J) 7→ eD (u, J, w, z) at a point (u, J, w, z) ∈ e−1
D (∆N )
is the map
N
M
T(u,J) M(A1 , . . . , AN , J ) →
Txν M ⊕ Txν M
ν=2
which assigns to the (N + 1)-tuple (ξ1 , . . . , ξN , Y ) the (2N − 2)-tuple (b
x, yb ) =
(b
x2 , . . . , x
bN , yb2 , . . . , ybN ) given by
x
bν = ξjν (wν ),
ybν = ξν (zν ).
Here we denote xν = ujν (wν ) = uν (zν ) and so x
bν , ybν ∈ Txν M . Given ybν it
follows from Lemma 6.1.2 that there exists a Y ∈ C ∞ (End(T M, J, ω)) and a
ξν ∈ C ∞ (u∗ν T M ) such that
ξν (zν ) = ybν .
Moreover, ξν can be chosen with support in a small small neighbourhood Bν of
zν and Y can be chosen with support in an arbitrarily small neighbourhood of
78
uν (Bν0 − Bν ) where Bν0 − Bν is a small annulus centered at zν By Proposition 2.3.2
these annuli can be chosen such that uν (Bν0 − Bν ) does not intersect any of the
other J-holomorphic curves uj (CP 1 ) for j 6= ν. Moreover, if wν 6= zjν then the ball
/ Bj0 ν and hence
Bj0 ν can be chosen so small that wν ∈
wν 6= zjν
=⇒
x
bν = ξjν (wν ) = 0.
If on the other hand wν = zjν then we have
wν = zjν
=⇒
x
bν = ξjν (wν ) = ybjν .
Now it is a simple combinatorial exercise that the space of all (2N − 2)-tuples
(b
x2 , . . . , x
bN , yb2 , . . . , ybN ) with arbitrary ybν and x
bν given by the above conditions is
transverse to the diagonal. This proves the proposition.
2
Proof of Theorem 6.3.2: The proof is essentially an application of the SardSmale theorem. As always, this theorem applies only to maps between Banach
manifolds and so one must work with the space J ` of almost complex structures of
class C ` and a suitable Sobolev space of J-holomorphic curves, as in the proof of
Theorem 3.1.2. These details will be left to the reader from now on.
Consider the projection
πD : M(D, J ) → J .
We first show that this map is Fredholm. To see this note that πD is a composite
M(D, J ) → M(A1 , J ) → J ,
where the second map πA1 is Fredholm. Therefore, it suffices to consider the first
map. But this has derivative
(ξ, Y, w,
b zb ) 7→ (ξ1 , Y ),
j
where (ξj , Y ) ∈ T(uj ,J) M(A , J ) for j = 1, . . . , N . Hence its kernel and cokernel
are finite dimensional. Therefore πD is Fredholm, and so its fibers M(D, J) are
manifolds for generic J.
We shall now prove that (u, J, w, z) is a regular point for πD if and only if at
this point the restricted evaluation map
eD : M(A1 , . . . , AN , J) × (CP 1 )2N −2 → M 2N −2
is transverse to the diagonal. But that (u, J, w, z) is a regular point for πD :
M(D, J ) → J just means that for every Y ∈ TJ J there exists (ξ, w,
b zb ) such that
(ξ, Y, w,
b zb) ∈ T(u,J,w,z) M(D, J ) or equivalently
deD (u, J, w, z)(ξ, Y, w,
b zb ) ∈ TeD (u,J,w,z) ∆N .
Since, by Proposition 6.3.3, the vectors
deD (u, J, w, z)(ξ, Y, w,
b zb )
j
with ξj ∈ Tuj M(A , J) are transverse to the multi-diagonal ∆N , it is easy to check
that the subspace of all such vectors with Y = 0 must already be transverse to ∆N .
Thus we have proved that J is a regular value for the projection πD : M(D, J ) → J
if and only if eD : M(A1 , . . . , AN , J) × (CP 1 )2N −2 → M 2N −2 is transverse to the
diagonal. This proves the theorem.
2
6.4. EVALUATION MAPS FOR CUSP-CURVES
6.4
79
Evaluation maps for cusp-curves
Note that the dimension of the moduli space M(D, J) with A1 + · · · + AN = A
is 2c1 (A) + 2n + 4(N − 1) and hence is equal to that of M(A, J) × (CP 1 )2N −2 .
However, this is deceptive, since the 6N -dimensional group GN = G × · · · × G acts
on this space by
φ · (uj , wν , zν ) = (uj ◦ φ−1
j , φjν (wν ), φν (zν )).
The space of simple unparametrized curves of type D is denoted by
C(D, J) = M(D, J)/GN .
By Proposition 2.3.2 the group GN acts freely on this space. Hence the quotient is
again a manifold for J ∈ Jreg (D) and its dimension is
dim C(D, J) = 2c1 (A1 + · · · + AN ) + 2n − 2N − 4.
Note that in the case N = 1 and A1 = A this is precisely the dimension of the space
C(A, J) = M(A, J)/G of unparametrized simple A-curves, namely 2c1 (A) + 2n − 6.
In general, however the dimension of the space C(D, J) will be at least two less
than that of C(A, J) and this observation lies at the heart of Theorem 5.2.1.
In order to define the domain W(D, J) of the correponding evaluation map we
must quotient M(D, J) out by as large a reparametrization group as we can in
order to reduce the dimension as much as possible. However, one component must
be kept parametrized so that it can be used to define an evaluation map. Therefore
fix ` ≤ N and let G`,N be the subgroup
G`,N = {φ = (φ1 , . . . , φN ) | φ` = id} ⊂ GN .
Then the space
C` (D, J) = M(D, J)/G`,N .
consists of all simplified cusp-curves C of type D whose components, except for the
`-th, are unparametrized. It follows that G acts on C` (D, J) by acting on its `-th
component in the usual way and so we define
W` (D, J) = C` (D, J) ×G CP 1
as the quotient space under the obvious action and
W(D, J) =
[
W` (D, J).
`
Again it follows from Proposition 2.3.2 that G acts freely on C` (D, J) × CP 1 and
hence the quotient space W` (D, J) is a manifold. There is, of course, an evaluation
map e : W(D, J) → M which is defined on W` (D, J) by
e(uj , wν , zν , ζ) = u` (ζ).
80
Lemma 6.4.1 Assume that A ∈ H2 (M, Z) is not a multiple class mB where m > 1
and c1 (B) = 0. Let D be the type of a simplified cusp-curve which represents the
class A. Suppose that J ∈ J + (M, ω, K) ∩ Jreg (D) is K-semi-positive for some
K > ω(A). Then W(D, J) is a manifold of dimension
dim W(D, J) ≤ dim W(A, J) − 2 max{1, (N − 1)}
for every D 6= A.
Proof: It just remains to check the statement about the dimension. First we have
dim C` (D, J)
=
dim M(A1 , . . . , AN , J) + dim (CP 1 )2N −2
− dim G`,N − codim ∆N
X
=
2c1 (Aj ) + 2nN
j
+4(N − 1) − 6(N − 1) − 2n(N − 1)
X
= 2n +
2c1 (Aj ) − 2(N − 1).
j
Now our hypotheses imply that either N > 1 or D = {A1 } where A = mA1 and
c1 (A1 ) > 0. In the first case, because c1 (Aj ) ≥ 0,
X
c1 (Aj ) ≤ c1 (A).
j
Thus, in either case,
dim W(D, J)
=
2n +
X
2c1 (Aj ) − 2(N + 1)
j
≤ 2n + 2c1 (A) − 4 − 2 max{1, (N − 1)}
=
dim W(A, J) − 2 max{1, (N − 1)}.
2
This proves the lemma.
Example 6.4.2 In the case of a cusp-curve with two components the description
of the compactified moduli space can be slightly simplified. In this case we may fix
a point z0 ∈ CP 1 and define
M0 (A, B, J)
= {(u, v) | u ∈ M(A, J), v ∈ M(B, J), u(z0 ) = v(z0 ),
u 6= v ◦ φ}
For generic J this space is a manifold of dimension 2c1 (A + B) + 2n. The 8dimensional group G0 × G0 acts on M0 (A, B, J) where G0 = {φ ∈ G | φ(z0 ) = z0 }.
Thus the space
M0 (A, B, J)
C(A, B, J) =
G0 × G0
of unparametrized cusp-curves of type (A, B) has dimension 2c1 (A + B) + 2n − 8,
which is 2 less than the dimension of the space C(A + B, J) = M(A + B, J)/G of
6.5. PROOFS OF THE THEOREMS IN SECTIONS 5.2 AND 5.3
81
unparametrized curves of type A + B. Now the natural domain of the evaluation
map can be defined as
W0 (A, B, J) =
M0 (A, B, J) M0 (A, B, J)
∪
G01 × G0
G0 × G01
where G01 = {φ ∈ G0 | φ(z1 ) = z1 } for some z1 6= z0 . Both groups G01 × G0 and
G0 × G01 act freely on M0 (A, B, J) and hence
dim W0 (A, B, J) = 2c1 (A + B) + 2n − 6.
Again this is 2 less than the dimension of M(A+B, J)×G CP 1 . The evaluation map
W → M is given by (u, v) 7→ u(z1 ) on the first component, and by (u, v) 7→ v(z1 )
on the second. It is easy to check that the space W0 (A, B, J) is diffeomorphic to
the above space W(D, J) for D = {A, B}.
2
6.5
Proofs of the theorems in Sections 5.2 and 5.3
We are now in a position to prove Theorems 5.2.1 and 5.3.1 on compactifying the
image of evaluation maps.
Proof of Theorem 5.2.1: By Gromov’s compactness theorem 4.4.3, the closure
of the set X(A, J) will contain points in M which lie on some cusp-curve which
represents the class A. Clearly, any such curve may be simplified, without changing
the set of points which lie on it, by deleting any repeated components and replacing
multiply covered components by their underlying simple components. Because
cusp-curves are connected the resulting simplified cusp-curve will lie in some set
C(D, J).
Thus, in order to compactify X(A, J) it suffices to add the points in X(D, J),
as D ranges over all effective, framed classes which can represent a simplified Jholomorphic cusp-curves whose energy is bounded by ω(A). By Corollary 4.3.3,
there are only finitely many of these. This corollary also implies that the set of
D which must be considered for each J is locally constant as J varies in J . This
proves (i). Statements (ii) and (iii) follow from Lemma 6.4.1. The important
observation here is that c1 (Aj ) ≥ 0 for all j and therefore simplification of a cuspcurve decreases the Chern number and hence the dimension of the space W(D, J).
2
Proof of Theorem 5.3.1: This is essentially the same as the proof of Theorem 5.2.1. The only challenge is to define the manifolds W(D, T, J, p) correctly.
Recall that the map
T : {1, . . . , p} → {1, . . . , N }
describes which of the N components of C are evaluated to obtain a p-tuple in M p .
Hence we define
X(D, T, J, p) = (x1 , . . . , xp ) | xj ∈ CT (j) , C ∈ C(D, J) ,
and set
W(D, T, J, p) = M(D, J) ×GN (CP 1 )p ,
82
where the ν-th factor φν of of φ = (φ1 , . . . , φN ) ∈ GN acts on M(D, J) as before,
and acts on the j-th factor of CP 1 if and only if T (j) = ν. Then there is an
evaluation map
eD,T : W(D, T, J, p) → M p
defined by
eD,T (u, J, w, z, ζ) = (uT (1) (ζ1 ), . . . , uT (p) (ζp ))
for (u, J, w, p) ∈ M(D, J) and ζ = (ζ1 , . . . , ζp ) ∈ (CP 1 )p . Now the dimension of
W(D, T, J, p) is given by
dim W(D, T, J, p)
=
dim M(D, J) + 2p − 6N
=
2
N
X
c1 (Aj ) + 2n + 4(N − 1) + 2p − 6N
j=1
=
2
N
X
c1 (Aj ) + 2n + 2p − 2N − 4.
j=1
In the case N = 1 with D = {A} this number agrees with the dimension of the
domain of the p-fold evaluation map on the space of simple curves
dim M(A, J) ×G (CP 1 )p = 2c1 (A) + 2n + 2p − 6.
But in the case N ≥ 2 or N = 1 with A = mA1 for m ≥ 2 the the dimension of
W(D, T, J, p) is at least two less than 2c1 (A) + 2n + 2p − 6. In view of this all the
statements in Theorem 5.3.1 follow from the corresponding statements in the case
p = 1.
2
The following lemma gives some useful information about the top dimensional
pieces in the boundary ∂C(A, J) = C(A, J) − C(A, J) of the moduli space of simple
curves.
Lemma 6.5.1 Assume that A ∈ H2 (M, Z) is not a multiple class mB where m > 1
and c1 (B) = 0 and suppose that J ∈ J + (M, ω, K) ∩ Jreg (D) is K-semi-positive for
some K > ω(A). Then the codimension 2 pieces in ∂C(A, J) consist of curves of
the following types:
(i) simple cusp-curves of type D = {A1 , A2 } where c1 (A1 + A2 ) = c1 (A);
(ii) simple B-curves where c1 (A) = 2 and A = 2B.
Proof: This follows from Lemma 6.4.1. Note that in (i) we must have m1 A1 +
m2 A2 = A for some m1 , m2 > 0, and hence we will have mi = 1 unless c1 (C i ) = 0.
2
6.6
Proof of the theorem in Section 5.4
The definition of the spaces V(D, T, J, z) which stratify the boundary of the set
Y (A, J, p) is slightly more complicated than that of the spaces W(D, T, J, p), because we must take into account the interaction of the points zi with the limiting
process.
6.6. PROOF OF THE THEOREM IN SECTION 5.4
83
There are four cases for the possible limit behaviour of the points ez (uν ) for a
sequence uν ∈ M(A, J). These are that uν converges modulo bubbling
(a) to a simple J-holomorphic sphere,
(b) to an m-fold covering of a simple J-holomorphic sphere where 2m < p,
(c) to an m-fold covering of a simple J-holomorphic sphere where 2m ≥ p,
(d) to a constant map.
We shall treat the cases (a) and (b) simultaneously. In fact, (a) is just the
special case m = 1. Let X ⊂ CP 1 be the finite set at which holomorphic spheres
bubble off. Then uν converges on the complement of X to a curve v 0 ◦ ψ where
v 0 ∈ M(A0 , J)
is a simple J-holomorphic curve representing the class A0 and ψ : CP 1 → CP 1 is
a rational map of degree m. Denote the space of such maps by Ratm . It has real
dimension
dim Ratm = 4m + 2.
Suppose that ` of the points zj lie in the set X and assume without loss of generality
that these are the points z1 , . . . , z` . Then we must consider limit curves of the form
(ψ, v 0 , . . . , v N ) with
v ν (0) = v 0 (ψ(zν )),
ν = 1, . . . , `.
These equations express the fact that the first ` bubbles are attached at the images
under v 0 ◦ ψ of the points z1 , . . . , z` . The remaining intersection pattern of the limiting curve is unconstrained, and can therefore be described by integers j`+1 , . . . , jN
with 0 ≤ jν ≤ ν − 1 such that
v ν (0) = v jν (wν ),
ν = ` + 1, . . . , N,
for some points wν ∈ CP 1 . Thus the intersection pattern of this curve can be coded
in a framing D of the form
D = {m, A0 , . . . , An , j`+1 , . . . , jN },
where there are positive integers m0 , m1 , . . . , mN such that m0 = m and
N
X
mj Aj = A.
j=0
There is a corresponding moduli space M(D, J, z) which consists of all 2N +2-tuples
(ψ, v 0 , v 1 , . . . , v N , w`+1 , . . . , wN )
satisfying these conditions, where v j represents the class Aj .
Lemma 6.6.1 For generic J, the moduli space M(D, J, z) is a manifold of dimension
dim M(D, J, z) = 2n + 2c1 (D) + 2N − 2` + 4m + 2.
84
Proof: Consider the evaluation map
eD : Ratm × M(A0 , . . . , AN , J ) × (CP 1 )N −` → M N × M N
defined by
eD (ψ, v 0 , . . . , v N , J, w`+1 , . . . , wN ) = (x, y)
with
x = (v 1 (0), . . . , v N (0))
and
y = (v 0 (ψ(z1 )), . . . , v 0 (ψ(z` )), v j`+1 (w`+1 ), . . . , v jN (wN )).
It can be proved as in Proposition 6.3.3 that this map is transverse to the diagonal
and hence the space M(D, J , z) = e−1
D (∆) is a Fréchet manifold.
Now consider the projection
M(D, J , z) → J .
As before the regular values of this projection are the points for which the restricted
evaluation map
eD,J : Ratm × M(A0 , . . . , AN , J) × (CP 1 )N −` → M N × M N
is transverse to the diagonal. For such values of J the space M(D, J, z) = e−1
D,J (∆)
is a manifold of dimension
dim M(D, J, z)
=
dim M(A0 , . . . , AN , J) + 2(N − `) + 4m + 2 − 2nN
=
2n(N + 1) + 2c1 (D) + 2(N − `) + 4m + 2 − 2nN
=
2n + 2c1 (D) + 2N − 2` + 4m + 2.
2
Now choose a map T : {1, . . . , `} → {1, . . . , N } and define the space
V(D, T, J, z) =
M(D, J, z) × (CP 1 )`
G × GN
0
where G0 = {φ ∈ G | φ(0) = 0}, and the group G × GN
0 acts by
φ · (ψ, v, w, ζ) = (φ0 ◦ ψ, v ν ◦ φ−1
ν , φjν (wν ), φT (ν) (ζν ))
for (ψ, v, w) ∈ M(D, J, z) and ζ ∈ (CP 1 )` . The evaluation map is defined by
eD,T,z (ψ, v, w, ζ) = (v T (1) (ζ1 ), . . . , v T (`) (ζ` ), v 0 (ψ(z`+1 )), . . . , v 0 (ψ(zN ))).
Note that the variables ζi parametrize the ith bubble, for i ≤ `. These bubbles
have to be added to the boundary of Y (A, J, p) because the set of points uν (zi )
might accumulate on any point on this bubble. Observe also that the image of the
points zj for j > ` is not determined completely by the map v 0 , but may be moved
around by difference choices of the multiple covering ψ : CP 1 → CP 1 .
85
Lemma 6.6.2 Under the hypotheses of Theorem 5.4.1(iii), the above manifolds
V(D, T, J, z) have dimension strictly less than that of M(A, J).
Proof: The group G0 is 4-dimensional and hence the space V(D, T, J, z) has
dimension
dim V(D, T, J, z)
=
dim M(D, z, J) + 2` − 4N
=
2n + 2c1 (D) − 2N + 4m − 4.
(We have added 2` to the dimension of M(D, J, z) to account for (CP 1 )` and
subtracted 4N + 6 for the dimension of G × GN
0 .) If m = 1 then we must have
N ≥ 1 and in this case the dimension is at most 2n + 2c1 (A) − 2 as claimed. Here
we only need the condition c1 (D) ≤ c1 (A) which is guaranteed by semi-positivity.
In fact,
N
X
c1 (Aj ) ≤ c1 (A) − (m − 1)c1 (A0 ).
c1 (D) =
j=0
Now suppose m > 1. Then by assumption of case (b) we have p > 2m and so
condition (JAp ) implies that c1 (A0 ) > 2. Hence
c1 (D) + 2m − 2 < c1 (A).
This implies again that
dim V(D, T, J, z) < 2n + 2c1 (A) = dim M(A, J).
Now we cannot assume in general that precisely the first ` points appear on the
bubbles. So for the general case we have to allow for a permutation of the points
z1 , . . . , zp and compensate with the inverse permutation of the factors in M p . Obviously such a permutation will not change the dimension count.
2
This takes care of limit points of the form (a) or (b). Now consider the case (c).
This means that uν converges to a multiply covered J-holomorphic sphere of class
mA0 where
p ≤ 2m.
The moduli space of corresponding simplified cusp-curves is determined by a framed
class
D = {A0 , . . . , AN , j1 , . . . , jN }
with (N + 1) components, and Chern number
c1 (D) ≤ c1 (A) − (m − 1)c1 (A0 ).
Since p ≤ 2m, the reparametrization map ψ acts transitively on p-tuples z of distinct points, and so the set of points uν (z) may accumulate anywhere on the limiting cusp-curve. Therefore, we must define the corresponding space V(D, T, J, z)
86
to be the space W(D, T, J, p) considered in the previous section. In view of Theorem 5.3.1 (iii) (with N replaced by N + 1) this space has dimension
dim W(D, T, J, p)
=
2n + 2c1 (D) + 2p − 2N − 6
≤ 2n + 2c1 (A) − 2(m − 1)c1 (A0 ) + 2p − 2N − 6
≤ 2n + 2c1 (A) − 2(m − 1)c1 (A0 ) + 4m − 2N − 6
=
2n + 2c1 (A) − 2(m − 1)(c1 (A0 ) − 2) − 2N − 2
≤ 2n + 2c1 (A) − 2
=
dim M(A, J) − 2.
Here we have used c1 (A0 ) ≥ 2 which follows from (JAp ).
To complete the proof that these strata have the properties required by Theorem 5.4.1(iii), it remains to consider the special cases p = 3, 4 with the weakened
assumptions stated there. Now the case p = 3 is covered by Theorem 5.3.1. Further, when p = 4 we must have m ≥ 2. Hence we may assume that at least one
of the components of the limit curve is multiply covered. Since, by (JA4 ), every
J-effective class B satisfies c1 (B) ≥ 1 this implies that
c1 (D) + 1 ≤ c1 (A).
If N ≥ 2 we obtain dim W(D, T, J, 4) < dim M(A, J) as required. The only other
case is N = 1. But then D = {B} where A = mB and so it follows from (JA4 )
that c1 (B) ≥ 2. This implies
c1 (D) + 2 ≤ c1 (A)
and with N = 1 we obtain again that dim W(D, T, J, 4) < dim M(A, J).
This covers the case (c). The case (d) occurs when the sequence uν (z) converges
to a constant J-holomorphic curve on the complement of a finite set X. This case
can be incorporated in the above treatment of the cases (a) and (b). Just take the
special case where m = 1 and A0 ∈ H2 (M, Z) is the zero homology class. It is a
simple matter to check that in this case we may remove the curve v 0 and instead
consider all tuples (v 1 , . . . , v N , w`+1 , . . . , wN ) which satisfy
v 1 (0) = · · · = v ` (0)
and
v ν (0) = v jν (wν ),
` + 1 ≤ ν ≤ N.
The corresponding domain of the evaluation map has dimension 2n + 2c1 (D) − 2N
and since N ≥ 1 this proves the required inequality. Here we only need to assume
that J is K-semi-positive for some K > ω(A) in order to ensure the inequality
c1 (D) ≤ c1 (A). This completes the proof of Theorem 5.4.1.
Remark 6.6.3 The assumption that there are no J-holomorphic curves with negative Chern number plays a crucial role in the proof of the dimension formula
in statement (iii) of all three Theorems 5.2.1, 5.3.1, and 5.4.1. If there is a Jholomorphic curve of class B with c1 (B) < 0 then an m-fold B-curve C 1 may
appear in the limiting cusp-curve C = C 1 ∪ · · · ∪ C N . Then simplifying the curve
87
C 1 would increase the dimension of the moduli space by 2(m − 1)|c1 (B)|. Hence
the codimension argument will fail, unless one can prove that such multiply covered curves with negative Chern number cannot appear as components of limiting
cusp-curves. So far no techniques have been developed to prove such statements for
(nonintegrable) almost complex structures. Similar results will also be needed to
remove the assumption (JAp ) in Theorem 5.4.1. The development of such results
would constitute an important breakthrough in extending the symplectic invariants
discussed in this paper (as well as Floer homology) to symplectic manifolds which
are not weakly monotone.
2
88
Chapter 7
Gromov-Witten Invariants
The Gromov-Witten invariants count the number of isolated curves which intersect
specified homology cycles in M . (Here a curve is called “isolated” if the dimension
of the appropriate moduli space is zero.) In the case of Φ, one does not care where
the curve intersects the cycles, while in defining Ψ these intersection points are
fixed in the domain S 2 .
Thus if M(A, J) denotes the moduli space of all J-holomorphic maps
u : CP 1 → M
which represent the homology class A ∈ H2 (M ; Z), consider the evaluation map
ep : M(A, J) ×G (CP 1 )p → M p
given by
ep (u, z1 , . . . , zp ) = (u(z1 ), . . . , u(zp ))
which was discussed in Chapter 1. The invariant Φ counts the number of intersection points of the image of this map with a d-dimensional homology class α in M p ,
where the dimension d is chosen so that, if all intersections were transverse, there
would be a finite number of such points. To define Ψ, one takes a generic point
z ∈ (CP 1 )p , where p ≥ 3, and counts the number of points in which the image of
M(A, J) × z meets the d-dimensional class α, where, again, d is chosen to make
this number finite.
Since the moduli space M(A, J) is not in general compact, we must show that
these numbers are well-defined, and independent of the choice of J and of representing cycle for α. To do this, it is convenient to introduce the notion of a pseudo-cycle.
We shall see that the compactness theorems of Chapter 5 may be rephrased by saying that the evaluation map ep , and its variant ez , represent pseudo-cycles in M p .
Then Φ and Ψ are simply the intersection numbers of these pseudo-cycles with
homology classes of complementary dimension in M p .
These invariants were first introduced into symplectic geometry in this generality
e Special cases of
by Ruan (cf. [64] and [65]). (Ruan denoted the invariant Ψ by Φ)
the invariant Φ were used by Gromov in [26] and by McDuff in [46] in order to get
information on the structure of symplectic manifolds. The invariant Ψ arises in the
context of sigma models and was considered by Witten in [85]. It is also possible to
89
90
CHAPTER 7. GROMOV-WITTEN INVARIANTS
define “mixed” invariants, in which one keeps track of the markings of only some of
the points (cf. [67]). In the terminology of Kontsevich and Manin [35] the invariant
Φ corresponds to the codimension-0 classes and the invariant Ψ to the classes of
highest codimension while the mixed invariants constitute intermediate cases.
7.1
Pseudo-cycles
Let M be a smooth compact m-dimensional manifold. An arbitary subset B ⊂ M
is said to be of dimension at most m if it can be covered by the image of a map
g : W → M which is defined on a manifold W of dimension m.1 In this case we
write dim B ≤ m. A k-dimensional pseudo-cycle in M is a smooth map
f :V →M
defined on an oriented k-dimensional manifold V such that roughly speaking, the
boundary of f (V ) is of dimension at most k − 2. More precisely, this boundary
has to be defined as the set of all limit points of sequences f (xν ) where xν has no
convergent subsequence in V . This agrees with the notion of the omega-limit-set
in dynamical systems, and we shall denote this set by
\
f (V − K).
Ωf =
K⊂V
K compact
Note that this set is always compact. Note also that if V is the interior of a
compact manifold V̄ with boundary ∂ V̄ = ∂V and f extends to a continuous map
f : V̄ → M then Ωf = f (∂V ). The condition on f to be a pseudo-cycle is
dim Ωf ≤ dim V − 2.
Two k-dimensional pseudo-cycles f0 : V0 → M and f1 : V1 → M are said to be
bordant if there exists a (k + 1)-dimensional oriented manifold W with ∂W =
V1 − V0 and a smooth map F : W → M such that
F |V0 = f0 ,
F |V1 = f1 ,
dim ΩF ≤ k − 1.
Pseudo-cycles in M form an abelian group with addition given by disjoint union.
The neutral element is the empty map defined on the empty manifold V = ∅. The
inverse of f : V → M is given by reversing the orientation of V . One could define
pseudo-homology of M as the quotient group of pseudo-cycles modulo the bordism
equivalence relation. It is not clear, however, whether the resulting homology groups
agree with singular homology. We shall return to this question at the end of this
section.
Remark 7.1.1 In order to represent a d-dimensional singular homology class α by
a pseudo-cycle f : V → M represent it first by a map f : P → M defined on a ddimensional finite oriented simplicial complex P without boundary. This condition
means that the oriented faces of its top-dimensional simplices cancel each other out
1 All our manifolds are σ-compact. This means that they can be covered by countably many
compact sets.
7.1. PSEUDO-CYCLES
91
in pairs. Thus P carries a fundamental homology class [P ] of dimension d and α is
by definition the class α = f∗ [P ]. Now approximate f by a map which is smooth
on each simplex. Finally, consider the union of the d and (d − 1)-dimensional faces
of P as a smooth d-dimensional manifold V and approximate f by a map which is
smooth across the (d − 1)-dimensional simplices.
Of course, things become easier if we work with rational homology H∗ (M, Q).
Because rational homology tensored with the oriented bordism group of a point is
isomorphic to rational bordism Ω∗ (M ) ⊗ Q, there is a basis of H∗ (M, Q) consisting
of elements which are represented by smooth manifolds. Thus we may suppose that
P is a smooth manifold, if we wish.
2
Exercise 7.1.2 Let M be compact smooth manifold of dimension dim M ≥ 3.
Prove that every 2-dimensional integral homology class A ∈ H2 (M, Z) can be represented by an oriented embedded surface. Hint: Note first that A can be represented by a finite cycle, because M can be triangulated. Every such cycle can
be thought of as a continuous map defined on a compact 2-dimensional simplicial
complex without boundary. Every such complex can be given the structure of a
smooth 2-dimensional compact manifold without boundary (which in the case of
integer coefficients is orientable). Hence A is represented by a continuous map
f : Σ → X defined on a smooth compact 2-manifold Σ. Approximate f by a
smooth map and use a general position argument to make f an immersion with
finitely many transverse self-intersections. In the cases dim M = 3 or dim M = 4
use a local surgery argument to remove the self-intersections. If dim M > 4 use a
general position argument to obtain an embedding.
2
Two pseudo-cycles e : U → M and f : V → M are called transverse if
Ωe ∩ f (V ) = ∅,
e(U ) ∩ Ωf = ∅,
and if
Tx M = im de(u) + im df (v)
whenever e(u) = f (v) = x. If e and f are transverse then the set
{(u, v) ∈ U × V | e(u) = f (v)}
is a compact manifold of dimension dim U + dim V − dim M . In particular, this
set is finite if U and V are of complementary dimension.
Lemma 7.1.3 Let e : U → M and f : V → M be pseudo-cycles of complementary
dimension.
(i) There exists a set Diff reg (M, e, f ) ⊂ Diff(M ) of the second category such that e
is transverse to φ ◦ f for all φ ∈ Diff reg (M, e, f ).
(ii) If e is transverse to f then the set {(u, v) ∈ U × V | e(u) = f (v)} is finite. In
this case define
X
e·f =
ν(u, v)
u∈U, v∈V
e(u)=f (v)
where ν(u, v) is the intersection number of e(U ) and f (V ) at the point e(u) =
f (v).
92
(iii) The intersection number e · f depends only on the bordism classes of e and f .
Proof: This is proved by standard arguments in differential topology as in [54]
and [28]. Here are the main points. The first statement can be proved by the same
techniques as the results of Chapter 6 with parameter space Diff(M ) instead of J .
It is easy to see that the map
U × V × Diff(M ) → M × M : (u, v, φ) 7→ (e(u), φ(f (v)))
is transverse to the diagonal and hence the universal space
M = {(u, v, φ) | e(u) = φ(f (v))} ⊂ U × V × Diff(M )}
is a manifold. Now consider the regular values of the projection M → Diff(M )
to obtain that e and φ ◦ f are transverse for generic φ. Now choose smooth maps
e0 : U0 → M and f0 : V0 → M such that dim U0 = dim U − 2 and Ωe ⊂ e0 (U0 )
and similarly for f . Apply the same argument as above to the pairs (e0 , f ), (e, f0 ),
(e0 , f0 ) to conclude that e(U ) ∩ φ(Ωf ) = ∅ and Ωe ∩ φ(f (V )) = ∅ for generic φ.
This proves (i).
Statement (ii) is obvious. To prove statement (iii) assume that the pseudo-cycle
f : V → M is bordant to the empty set with corresponding bordism F : W → M .
As in (i) it can be proved that this bordism can be chosen transverse to e and,
moreover,
Ωe ∩ F (W ) = ∅,
e(U ) ∩ ΩF = ∅.
Hence the set
X = {(u, v) ∈ U × W, | e(u) = F (v)}
is a compact oriented 1-manifold with boundary
∂X = {(u, v) ∈ U × V | e(u) = f (v)} .
2
This proves that e · f = 0.
Every (m − d)-dimensional pseudo-cycle e : W → M determines a homomorphism
Φe : Hd (M, Z) → Z
as follows. Represent the class β ∈ Hd (M, Z) by a cycle f : V → M as in Remark 7.1.1. Any two such representations are bordant and hence, by Lemma 7.1.3,
the intersection number
Φe (β) = e · f
(7.1)
is independent of the choice of the cycle f representing β. The next assertion also
follows from Lemma 7.1.3.
Lemma 7.1.4 The homomorphism Φe depends only on the bordism class of e.
Thus we have proved that every (m − d)-dimensional pseudo-cycle e : U → M
determines a homomorphism Φe : Hd (M, Z) → Z and hence an element of the space
H d (M ) = H d (M, Z)/torsion = Hom(Hd (M, Z), Z) which we denote by
ae = [Φe ] ∈ H d (M ).
7.2. THE INVARIANT Φ
93
Thus we have defined a map from bordism classes of pseudo-cycles to H d (M ). If e
is actually a smooth cycle (as in Remark 7.1.1) then the homology class of e is the
Poincaré dual of ae = PD([e]). In general, one can think of e as a weak representative
of the class αe = PD(ae ) ∈ Hm−d (M ). In other words, an (m − d)-dimensional
pseudo-cycle e : U → M is called a weak representative of the homology class
α ∈ Hm−d (M ) if e·f = α·β for every homology class β ∈ Hd (M, Z) and every cycle
f representing β (as in Remark 7.1.1). That this is a meaningful definition requires
the proof that the formula (7.1) continues to hold when f : V → M is only a
pseudo-cycle representing the class β in the weak sense just defined. This assertion
is not obvious because we do not know whether any two weak representatives of a
homology class β ∈ Hd (M ) are bordant.
Lemma 7.1.5 Let e : U → M be an (m − d)-dimensional pseudo-cycle. If the
d-dimensional pseudo-cycle f : V → M is a weak representative of the homology
class β ∈ Hd (M ) then Φe (β) = e · f .
Proof: It suffices to prove the assertion in the case β = 0. Hence assume that f has
intersection number 0 with every (m−d)-dimensional smooth cycle. We must prove
that f has intersection number 0 with every (m − d)-dimensional pseudo-cycle. To
see this assume that the (m − d)-dimensional pseudo-cycle e : U → M is in general
position. Thus e(U ) does not intersect the (compact) limit set Ωf . Now choose
a sufficiently small open neighbourhood W ⊂ M of Ωf with smooth boundary,
transverse to f . Then V0 = f −1 (M − W ) is a compact manifold with boundary
and the restriction of f to V0 is a smooth map f0 : (V0 , ∂V0 ) → (M − W, ∂W ). This
map has intersection number zero with every (m − d)-dimensional cycle in M − W .
Hence there is an integer ` > 0 such that the `-fold multiple of f0 is a boundary
in Hd (M − W, ∂W ; Z). This implies that the pseudo-cycle e : U → M − W has
intersection number zero with f0 and hence with f .
2
7.2
The invariant Φ
Let (M, ω) be a compact symplectic manifold and fix a class A ∈ H2 (M, Z). We
will assume throughout that (M, ω) and A satisfy the following hypotheses.
(H1) The class A is not a nontrivial multiple of a class B with c1 (B) = 0.
(H2) The manifold (M, ω) is weakly monotone.
Recall from Section 5.1 that a symplectic manifold (M, ω) is called weakly monotone
if every spherical homology class A ∈ H2 (M, Z) with ω(A) > 0 and c1 (A) < 0 must
satisfy c1 (A) ≤ 2 − n. This condition guarantees that, for generic J, there are no Jholomorphic spheres in homology classes of negative first Chern number. As we saw
in Chapters 5 and 6, this is the essential condition which makes the compactness
theorems work.
Given a spherical homology class A ∈ H2 (M, Z) we shall define homomorphisms
ΦA,p : Hd (M p , Z) → Z
for p ≥ 1 where
d = 2(n − 1)(p − 1) + 4 − 2c1 (A).
(7.2)
94
Roughly speaking, the number ΦA,p (α) counts (with multiplicities) the number of
J-holomorphic curves in the homology class A whose p-fold product intersects a
generic representative of the class α ∈ Hd (M p , Z). The condition (7.2) guarantees
that for a generic choice of the almost complex structure J there are finitely many
such curves. Mostly we shall evaluate ΦA,p on homology classes α ∈ Hd (M p , Z)
which are products of classes αj ∈ Hdj (M, Z) with d1 + · · · + dp = d and, for such
α, will denote ΦA,p (α) by
ΦA (α1 , . . . , αp )
suppressing the notation p. Thus we may think of ΦA,p as a collection of homomorphisms
Hd1 (M, Z) ⊗ · · · ⊗ Hdp (M, Z) → Z
where
d = d1 + · · · + dp
satisfies (7.2).
To define ΦA,p we fix a generic ω-tame almost complex structure J ∈ J (M, ω).
(More precisely, it should be regular in the sense of Theorem 5.3.1. See also Remark 5.1.1.) By Theorem 3.1.2, the space M(A, J) of J-holomorphic curves has
dimension 2n + 2c1 (A). Further, formula (7.2) shows that the domain
W(A, J, p) = M(A, J) ×G (CP 1 )p
of the p-fold evaluation map ep has dimension
dim W(A, J, p)
=
2n + 2c1 (A) + 2p − 6
=
2np − d
p
X
(2n − di ).
=
i=1
where d1 + · · · + dp = d. Choose homology classes αi ∈ Hdi (M ; Z) for 1 ≤ i ≤ p.
Then the set of A-curves which intersect the classes α1 , . . . , αp is a finite set provided
that the cycles which represent these classes are in general position. The invariant
ΦA,p (α1 , . . . , αp ) is defined as the number of such curves, counted with appropriate
signs:
ΦA,p (α1 , . . . , αp ) = # {[u, z1 , . . . , zp ] ∈ W(A, J, p) | u(zi ) ∈ αi } .
(A more formal definition is given below.) We shall see that, in the weakly monotone
case, this invariant is independent of the choice of J. Moreover, the form ω is not
actually needed for the definition of the invariant ΦA,p except as a tool to prove
finiteness. It follows that ΦA,p does not change when ω varies, provided that it
remains weakly monotone.
Evaluation maps as pseudo-cycles
The assertion of Theorem 5.3.1 can be restated in the form that the evaluation map
ep = eA,J,p : W(A, J, p) = M(A, J) ×G (CP 1 )p → M p
95
defined by ep (u, z1 , . . . , zp ) = (u(z1 ), . . . , u(zp )) determines a pseudo-cycle whenever
J ∈ J+ (M, ω, K) for some K > ω(A) and J is regular. To see this observe that
the lower dimensional strata are given by the evaluation maps
eD,T : W(D, T, J, p) → M p .
All these manifolds W(D, T, J, p) are σ-countable and have dimension at least two
less than that of W(A, J, p). Moreover, the union of the sets eD,T (W(D, T, J, p))
cover the omega-limit-set ΩeA,J,p . Hence the map eA,J,p : W(A, J, p) → M p is a
pseudo-cycle as claimed.
It follows from the above discussion that this map determines a homomorphism
ΦA,J,p : Hd (M p , Z) → Z
with
d = 2(n − 1)(p − 1) + 4 − 2c1 (A),
(provided that J ∈ J+ (M, ω, K) for some K > ω(A) and J is regular). More
explicitly, represent the homology class α ∈ Hd (M p , Z) by a cycle f : V → M p
as in Remark 7.1.1. By Lemma 7.1.3 (i), put this map in general position so that
it becomes transverse to eA,J,p . By Lemma 7.1.3 (ii), this implies that the set
ep (W(A, J, p)) ∩ f (V ) is finite. Define
ΦA,J,p (α) = eA,J,p · f
as the oriented intersection number. Lemma 7.1.3 (iii) implies that the right hand
side depends only on the bordism class of f and hence only on the homology class
α. Moreover, Lemma 7.1.5 shows that the formula ΦA,J,p (α) = eA,J,p · f continues
to hold for every pseudo-cycle f : V → M p which is a weak representative of the
class α. By Lemma 7.1.4, the homomorphism ΦA,J,p : Hd (M p , Z) → Z depends
only on the bordism class of the evaluation map eA,J,p .
The invariant ΦA,J,p (α) vanishes on all the torsion elements α ∈ Hd (M p , Z).
Since the free part Hd (M p , Z)/Tor is generated by product cycles α = (α1 ×· · ·×αp ),
one often restricts ΦA,J,p to such cycles and writes
ΦA,J,p (α1 , . . . , αp ) = ΦA,J,p (α1 × · · · × αp ).
It is convenient to set ΦA,J,p (α1 , . . . , αp ) = 0 when the dimension condition (7.2)
is not satisfied.
Remark 7.2.1 It is interesting to note that the invariant ΦA,J,p (α1 , . . . , αp ) can
be defined as the intersection number of eA,J,p with a product cycle f1 × · · · × fp
where the fj : Vj → M are pseudo-cycles representing the homology classes αj ,
respectively. In other words, transversality can be achieved within the class of
product cycles. This is a simple adaptation of Lemma 7.1.3 (i) to the case where f
is a product cycle and φ is a product diffeomorphism.
2
Proposition 7.2.2 Assume that (M, ω) is weakly monotone and A is not a nontrivial multiple of a class B with c1 (B) = 0. Then the homomorphism
ΦA,p = ΦA,J,p : Hd (M p , Z) → Z
is independent of the choice of the regular ω-tame almost complex structure J ∈
Jreg (M, ω) used to define it.
96
Proof: To prove this we need a version of Theorem 5.3.1 which is valid for regular
homotopies {Jλ }. That is, we need to know that, given two regular almost complex
structures J0 and J1 , any path joining them may be perturbed so that the sets
[
W(D, T, {Jλ }λ , p) = {λ} × W(D, T, Jλ , p)
λ
are smooth manifolds for all D and T . This follows by the same argument which
proves Theorem 5.3.1. We conclude that the evaluation maps
eA,J0 ,p : W(A, J0 , p) → M p ,
eA,J1 ,p : W(A, J1 , p) → M p ,
determine bordant pseudo-cycles. The statement now follows from Lemma 7.1.4.
2
A deformation of a symplectic form ω is a smooth 1-parameter family of
ωt , t ∈ [0, 1] of forms starting at ω0 = ω. In distinction to the notion of isotopy,
we do not require that the cohomology class remain constant. Because the taming
condition is open, an almost complex structure J which is tamed by ω is tamed
by all sufficiently close symplectic forms. It follows easily that the homomorphism
ΦA,p does not change under a deformation ωt of ω, provided that (M, ωt ) is weakly
monotone for all t. Since manifolds of dimension ≤ 6 and manifolds where c1
vanishes on spherical classes are always weakly monotone, ΦA,p is a deformation
invariant in these cases.
For part of our discussion the condition (H2) can be replaced by the following
weaker hypothesis. Again, the relevant definitions come from Section 5.1.
(H3) The set J+ (M, ω, K) of all K-semi-positive almost complex structures which
are compatible with ω is nonempty for some K > ω(A).
If we replace the condition that (M, ω) be weakly monotone by the weaker hypothesis (H3) then the homomorphism ΦA,J,p : Hd (M p , Z) → Z can still be defined
for generic J ∈ J+ (M, ω, K) with K > ω(A). However, in this case the methods
in the proof of Proposition 7.2.2 only show that ΦA,J,p is locally independent of
J and we do not know in general whether or not the space J+ (M, ω, K) is path
connected. Thus, to prove that they are invariant, i.e. independent of the almost
complex structure J used to define them, we need hypothesis (H2).
Another situation in which there is a well-defined invariant is when M is a Kähler
manifold (M, J0 , ω) which satisfies one of the conditions mentioned in Remark 5.1.4.
We formulate this here for the invariant Φ, but clearly a similar result holds for Ψ.
Observe that here we somewhat change our perspective, thinking of the complex
structure J0 as given data rather than the symplectic form ω.
Proposition 7.2.3 Suppose that (M, ω, J0 ) is a Kähler manifold such that J0 is
tamed by a weakly monotone symplectic form ω 0 . Suppose also that A is not a
nontrivial multiple of a class B with c1 (B) = 0. Then there is a neighbourhood
N (J0 ) of J0 in the space of all almost complex structures on M such that the
homomorphism
ΦA,p = ΦA,J,p : Hd (M p , Z) → Z
97
is well-defined and independent of the choice of the regular element J ∈ N (J0 ) used
to define it, and of the taming form ω 0 . It depends on J0 up to deformations through
complex structures which are tamed by weakly monotone symplectic forms.
Proof: This is almost obvious. Since the taming condition is open, there is
a path-connected neighbourhood N (J0 ) of J0 in the space of all almost complex
structures on M , the elements of which are all tamed by ω 0 . Since ω 0 is monotone,
the evaluation map eA,J,p will define a pseudo-cycle for generic J ∈ N (J0 ). Just
as before, its homotopy class is independent of the choice of the generic element
J ∈ N (J0 ), and of the choice of neighbourhood N (J0 ). The rest of the proposition
is clear.
2
Remark 7.2.4 (Integration over moduli spaces) There is an alternative way
to express the invariant ΦA,p in terms of the integrals of certain differential forms
over the moduli space W(A, J, p). Namely, given a homology class α ∈ Hd (M p )
where d satisfies (7.2), we can define ΦA,J,p (α) as the integral
Z
ΦA,J,p (α) =
eA,J,p ∗ τ.
W(A,J,p)
Here the differential form τ is closed and represents the Poincaré dual a = PD(α) ∈
H 2n−d (M d ). The dimension condition (7.2) asserts that
deg τ = 2n + 2c1 (A) + 2p − 6 = dim W(A, J, p)
and so eA,J,p ∗ τ is a top-dimensional form on W(A, J, p). If one takes this approach
one must show firstly that the integral is finite, secondly that it is independent of
the differential form chosen to represent the class a, and thirdly as before that it
is independent of the almost complex structure J. The first problem can be solved
by choosing a differential form which is supported near the image of a generic
pseudo-cycle f : V → M p which represents the Poincaré dual α = PD(a). If f is
transverse to the evaluation map then the pullback eA,J,p ∗ τ has compact support.
This argument shows that the integral is finite for some form which represents the
class a. To prove this for all forms representing a, and to prove that the integral
is independent of the choice of the form, one must show that the pullback of any
exact form integrates to zero, that is
Z
eA,J,p ∗ dσ = 0
W(A,J,p)
for every form σ of degree dim W(A, J, p) − 1. Intuitively, this should be the case
because the boundary of W(A, J, p) is of codimension 2 and so the integral of σ over
it should vanish. However, to make this precise is somewhat nontrivial. In the case
of intersection theory this problem can be avoided because of Sard’s theorem. 2
In order to calculate ΦA,p , one has to be able to recognise when an almost
complex structure J is generic (i.e. in the appropriate set Jreg ). Our arguments
show that J is generic if it is regular for all classes Aj which can appear as components of a reducible A-curve, and is such that all the relevant evaluation maps
98
are transverse. For example, if J is regular for A-curves (which one can check by
using Lemma 3.5.1) and if A is a simple class, then there are no reducible A-curves
and J is generic in the required sense. In practice, one does not need all these
conditions to be satisfied. All that is required to make the above constructions
work is that J be “good”, in the sense that J is regular for A-curves and that all
the manifolds W(D, T, J, p) have dimension at least 2 less than that of W(A, J, p).
Then the evaluation map eA,J,p does define a pseudo-cycle in the correct homology
class.
Another important ingredient in the calculation of ΦA is the determination of
the correct signs. If J is integrable, we saw in Remark 3.3.6 that the moduli spaces
have complex structures which are compatible with their orientations. Hence in this
case it is usually easy to figure out the sign of intersection points (see Example 7.3.6
below). However, if J is not integrable, the question is much more delicate. When
M has (real) dimension 4 and one is considering (non-empty) moduli spaces of
spheres, McDuff in [47] shows that when c1 (A) = 1 + p ≥ 2 the evaluation map
ep : W(A, J, p) → M p
preserves orientation. The condition c1 (A) = 1 + p ensures that ep maps between
spaces of equal dimension, and so guarantees that ΦA,p (pt, . . . , pt) is defined. Thus
this result implies that whenever there is a J-holomorphic sphere with Chern number 1 + p ≥ 2 in a compact symplectic 4-manifold for any ω-tame J, the invariant
ΦA,p (pt, . . . , pt) > 0.
It follows that the moduli space M(A, J) is never empty for any ω-tame J. This
result has been proved only for curves of genus 0, and it is not clear what happens
for a general symplectic 4-manifold when the genus is greater than 0. For example,
the analogous result for tori would say that if there is a J-holomorphic torus with
Chern number at least 1 for one ω-tame J, then there is such a torus for all ω-tame
J. It is not known whether this holds. Some relevant examples can be found in
Lorek [41].
7.3
Examples
Example 7.3.1 (Lines in Projective space) In CP n two points lie on a unique
line. This should translate into the statement that
ΦL (pt, pt) = 1,
where L = [CP 1 ] ∈ H2 (CP n , Z) and pt ∈ H0 (CP n , Z) is the homology class of a
point.
To verify that this is indeed the case, note first that, since c1 (A) = n + 1 and
p = 2, the dimension equation (7.2) works out correctly with d = 0. Further, the
usual complex structure J0 satisfies the conditions of Lemma 3.5.1. In fact, because
CP 1 ⊂ CP 2 ⊂ CP 3 and so on, the normal bundle to a complex line is a sum of
line bundles each of Chern number 1. Therefore, J0 satisfies the transversality
requirement of Theorem 5.3.1 needed for the definition of ΦL . (All the evaluation
maps eD : M(A1 , . . . , AN ) × (CP 1 )2N −2 → M 2N −2 are transverse to the diagonal
7.3. EXAMPLES
99
∆N ⊂ M 2N −2 .) Hence ΦL (pt, pt) is exactly the number of lines through two points,
and so equals 1.
2
Example 7.3.2 (Conics in Projective space) In order to calculate the invariant ΦA (α1 , . . . , αp ) when A = 2L using the standard complex structure J0 one needs
to check not only that J0 is regular for the class 2A (which follows by Lemma 3.5.1)
but also that the space of cusp-curves of type D = (A, A) has small enough dimension. In fact, we need to check that the manifold W(D, J0 ) which was defined in
Chapter 5 above has dimension at least 2 less than dim W(2L, J0 ) = 6n. This is
an easy exercise. Thus, because there is a unique conic through 5 points in CP 2 ,
one finds that
Φ2L (pt, pt, pt, pt, pt) = 1
in CP 2 . Further, if the αi are all points then
Φ2L (α1 , . . . , αp ) = 0
n
in CP with n > 2. To see this, observe that when n = 3 one would need p = 4
points to make condition (7.2) hold, but there is no conic through non-coplanar 4
points. (In fact, any three points determine a unique 2-plane P . If C is a conic
through these three points then either C ⊂ P or the intersection number C · P ≥ 3.
But we know for homological reasons that C ·P = 2. Therefore C must be contained
in P , which means that we cannot choose the 4th point freely.) Further, if n > 3
it is impossible to choose p so that condition (7.2) holds. Thus, if all the αi are
points, the invariants are rather limited. But with other choices of αi one does get
non-trivial invariants. For example, it is not hard to check that in CP 3
Φ2L (pt, pt, pt, line, line) = 1.
2
Example 7.3.3 (Cylinders) The result of Example 1.5.1 can be rephrased as the
statement that
ΦA (pt) = 1,
where A = [CP 1 × pt] ∈ H2 (CP 1 × V ). By Proposition 7.2.2, this holds whenever
(CP 1 × V, ω) is weakly monotone, and in particular if π2 (V ) = 0 or dim V ≤ 4. 2
Example 7.3.4 (Rational and ruled symplectic 4-manifolds) Let (M, ω) be
a compact symplectic 4-manifold which contains a symplectically embedded 2sphere S with self-intersection number
S·S ≥0
but no such sphere with self-intersection number −1. Under these conditions it
is proved in [45] (see also [51]) that there must be a symplectically embedded 2sphere C of self-intersection number either 0 or 1. In the latter case the pair
(M, C) is symplectomorphic to (CP 2 , CP 1 ). In the former case C · C = 0 and
M is diffeomorphic to a ruled surface, i.e. an S 2 -bundle over a Riemann surface.
The base is the moduli space of J-holomorphic spheres representing the homology
class A = [C] and the fibers are the J-holomorphic curves themselves (compare
with the previous example). Positivity of intersection shows that these fibers are
mutually disjoint and the adjunction formula for singular curves shows that they
are embedded.
100
Example 7.3.5 (Exceptional divisors) Consider the manifold −CP 2 , which is
the complex projective plane with the reversed orientation. If A ∈ H2 (−CP 2 ; Z)
be the class of the complex line CP 1 then A · A = −1.
f denote its blow
Now suppose that M is a complex Kähler surface, and let M
up at the point x. This means that the point x is replaced by the set of all lines
through x. This set of lines is a copy of CP 1 which is called the exceptional
divisor Σ. The normal bundle to Σ is the canonical line bundle over CP 1 which
f is diffeomorphic to the connected sum
has Euler class −1, and it follows that M
2
M #(−CP ) by a diffeomorphism which identifies Σ with a complex line in −CP 2 .2
Let E = [Σ] denote the homology class of the exceptional divisor. Then the moduli
space M(E, J) consists of a single J-holomorphic curve, up to reparametrization,
namely the exceptional divisor itself. To see this note that any other J-holomorphic
curve in this class would have to intersect Σ with intersection number −1, but this
is impossible because the intersection number of any two distinct J-holomorphic
curves is nonegative. Note that, by Lemma 3.5.2, the curve Σ is regular, and this is
consistent with the dimension formula dim M(E, J) = 4 + 2c1 (E) = 6. Therefore
we find that
ΦE (E) = E · E = −1.
2
Example 7.3.6 (Non-deformation equivalent 6-manifolds) This example is
due to Ruan (cf. [65]). Let X be CP 2 with 8 points blown up, and Y be Barlow’s
surface. These manifolds are simply connected and have the same intersection form.
Hence they are homeomorphic, but they are not diffeomorphic. By results of Wall,
the stabilized manifolds X 0 = X × CP 1 and Y 0 = Y × CP 1 are diffeomorphic.
Moreover, each of them is Kähler, and one can choose the diffeomorphism
ψ : X0 → Y 0
so that it preserves the Chern classes of the respective complex structures. The
latter statement implies that ψ takes the complex structure JX on X 0 to a structure
ψ∗ (JX ) which is homotopic to the obvious complex structure JY on Y 0 . Moreover
an easy argument using cup-products shows that ψ∗ [X × pt] = [Y × pt], and this
can be used to show that any class A = [Σ × pt] ∈ H2 (X 0 ) is mapped to a class of
the form ψ∗ A = [Σ0 × pt] ∈ H2 (Y 0 ).
Now, on any Kähler manifold, the set of Kähler forms is a convex cone, and so
any two such forms are deformation equivalent. (In fact, ω and ω 0 can be joined
by the deformation tω + (1 − t)ω 0 .) In other words, a Kähler manifold Z carries a
natural deformation class of symplectic forms ωZ . It is natural to ask whether the
Kähler manifolds (X 0 , ωX 0 ) and (Y 0 , ωY 0 ) are deformation equivalent. Ruan shows
that they are not, by calculating
ΦE (E)
where E ∈ H2 (X 0 , Z) represents the class of one of the blown up points in X. Since
c1 (E) = p = 1, the dimension requirement
2c1 (E) = 2(n − 1)(p − 1) + 4 − dim α,
is satisfied. Further, since there is exactly one holomorphic sphere Σ in X which
represents E, the E-curves in X 0 = X × CP 1 are precisely the curves Σ × {z}, for
2 This
is all fully explained in Griffiths and Harris [25], or McDuff and Salamon [52].
7.4. THE INVARIANT Ψ
101
z ∈ CP 1 . Hence the set of points X(E, JE ) which lie on JX -holomorphic E-curves
is just Σ × CP 1 . Since E · E = −1 in X, it is easy to check that the intersection
number E · X(E, JX ) = −1, and hence
ΦE (E) = −1.
However, the Barlow surface Y is minimal and so has no holomorphic curves with
self-intersection number −1. Hence there are no JY -spheres in Y 0 = Y ×CP 1 which
represent the class ψ∗ E. Here we use the fact that ψ∗ E = [Σ0 × pt] and so any
JY -holomorphic representative of ψ∗ E would have to be of the form Σ0 × pt where
Σ0 · Σ0 = −1. Thus JY is generic in the requisite sense, and so Φψ∗ E vanishes on
Y 0 . But if ψ were a deformation equivalence then Φψ∗ E (ψ∗ E) would have to be
nonzero.
2
Example 7.3.7 (Counting discrete curves) If 2c1 (A) = 6 − 2n then the space
C(A, J) of unparametrized A-curves is discrete (i.e. has dimension 0) and one
might want to count the number N (A) of these curves, with appropriate signs
corresponding to the natural orientation of moduli space. These orientations can
be defined as in Chapter 3. (It makes no difference that the moduli space is of
dimension zero. Also, if J is integrable, it follows from Remark 3.3.6 that the
orientation of each point is positive though it need not be for general J. See [42]
and [64].) In order for condition (H1) to hold, we need c1 (A) > 0, which restricts
us to the case 2n = 4. In this case there is, by Poincaré duality, another class
α ∈ H2 (M ) such that A · α = k 6= 0, and it is easy to check that
ΦA (α) = kN (A).
Observe further, that if A is represented by an embedded sphere, the adjunction
formula implies that A·A = −1. As in Example 7.3.5, it then follows from positivity
of intersections that N (A) = 1. However, this need not be true in general.
When 2n = 6, there is a way to count the curves, avoiding the problems caused
by multiple covers, which we describe in Section 9.3 below. When 2n > 6 and
the space M(A, J) is nonempty discrete curves must have c1 (A) < 0. Therefore
condition (H1) cannot hold and the present methods do not treat this case.
2
7.4
The invariant Ψ
We assume in this section that a generic almost complex structure J on M and
the homology class A ∈ H2 (M ) satisfy the condition (JAp ) of Section 5.4. We will
consider the following additional conditions.
(H4) If A = mB is a nontrivial multiple of a homology class B with m > 1 then
either c1 (B) ≥ 3 or p ≤ 2m.
(H5) For a generic almost complex structure J ∈ J (M, ω) every J-effective homology class A ∈ H2 (M ) has Chern number c1 (A) ≥ 2.
102
Note that in the cases p = 3 and p = 4 we always have p ≤ 2m and so condition (H4)
is empty. Condition (H5) is satisfied, for example, if the manifold (M, ω) is weakly
monotone with minimal Chern number N ≥ 2. This implies that either (M, ω) is
monotone or the minimal Chern number is N ≥ n−2. However, the important case
of Calabi-Yau manifolds with Chern class c1 = 0 is excluded by these assumptions.
If (H4) and (H5) are satisfied we shall define a homomorphism
ΨA,p : Hd (M p , Z) → Z
for p ≥ 1 where
d = 2n(p − 1) − 2c1 (A).
(7.3)
Note that in the case p = 3 this number d agrees with the one given by (7.2). In
fact for p = 3 the invariant ΨA,p agrees with ΦA,p , but for p > 3 these invariants
are different. Roughly speaking, we shall fix a p-tuple z = (z1 , . . . , zp ) ∈ (CP 1 )p
and homology classes αj ∈ Hdj (M, Z) with
d1 + · · · + dp = d,
and define
ΨA,p (α1 , . . . , αp ) = # {u ∈ M(A, J) | u(zj ) ∈ αj } .
The condition (7.3) will guarantee that, for a generic almost complex structure J
and a generic point z ∈ (CP 1 )p , this is a finite set. The invariant ΨA,p is the
number of points in this set, counted with appropriate signs.
In order to make this precise we shall assume that J is regular in the sense
of Definition 3.1.1 so that the moduli space M(A, J) is a manifold of dimension
2n + 2c1 (A). Now consider the evaluation map
eA,J,z : M(A, J) → M p
defined by
eA,J,z (u) = (u(z1 ), . . . , u(zp )).
Theorem 5.4.1 shows that for generic J the closure of the image of this map can be
represented as the union of ez (M(A, J)) with lower dimensional strata. In other
words the map eA,J,z represents again a pseudo-cycle and therefore determines a
homomorphism
ΨA,J,z : Hd (M p , Z) → Z,
d = 2n(p − 1) − 2c1 (A).
As before, represent a homology class
α = α1 × · · · × αp ∈ Hd (M p , Z)
by a pseudo-cycle f = f1 × · · · × fp : V1 × · · · × Vp → M p (see Remark 7.1.1). By
Lemma 7.1.3 (i) put this map in general position so that it becomes transverse to
the maps eD,T,z for all D and T . By Lemma 7.1.3 (ii) this implies that the set
eA,J,z (M(A, J)) ∩ f (V ) is finite. Thus, we may define
ΨA,J,z (α1 , . . . , αp ) = eA,J,z · f
as the oriented intersection number. By Lemma 7.1.3 (iii) the right hand side
depends only on the bordism class of f and hence only on the homology class
α. Further, by Lemma 7.1.4, this map depends only on the bordism class of the
evaluation map eA,J,z .
103
Lemma 7.4.1 Assume (H4) and (H5). Then the homomorphism
ΨA,p = ΨA,J,z : Hd (M p , Z) → Z
is independent of the choice of the regular ω-tame almost complex structure J ∈
Jreg (M, ω) and the p-tuple z = (z1 , . . . , zp ) ∈ (CP 1 )p used to define it. Hence ΨA,p
depends only on the deformation class of ω.
Proof: As in the case of Proposition 7.2.2 the proof relies on a version of Theorem 5.4.1 for regular homotopies {Jλ } and {zλ }. We must show that any two
regular almost complex structures J0 and J1 and any two points z0 , z1 ∈ (CP 1 )p
can be joined by regular paths {Jλ } and {zλ } such that the spaces
[
V(D, T, {Jλ }, {zλ }) = {λ} × V(D, T, Jλ , zλ )
λ
are smooth manifolds for all D and T . This follows by the same arguments used in
the proof of Theorem 5.4.1. We conclude that the evaluation maps
eA,J0 ,z0 : M(A, J0 ) → M p ,
eA,J1 ,z1 : M(A, J1 ) → M p ,
determine bordant pseudo-cycles. The statement now follows from Lemma 7.1.4.
2
Remark 7.4.2 (Integration over moduli spaces) As in Remark 7.2.4 the invariant ΨA,J,z (α1 , . . . , αp ) can be represented as an integral of a differential form
over the moduli space M(A, J). Represent the cohomology classes aj = PD(αj ) ∈
H 2n−dj (M ) by closed forms τj ∈ Ω2n−dj (M ) and define
Z
ΨA,J,z (α1 , . . . , αp ) =
e∗1 τ1 ∧ e∗2 τ2 ∧ · · · ∧ e∗p τp
M(A,J)
where ej : M(A, J) → M denotes the evaluation map ej (u) = u(zj ). The dimension
condition (7.3) means that
p
X
deg τj = 2n + 2c1 (A) = dim M(A, J)
j=1
and so the exterior product of the forms e∗j τj is a top-dimensional form on M(A, J).
The difficulties in making this approach rigorous are the same as those discussed
in Remark 7.2.4.
2
Remark 7.4.3 Both invariants ΦA and ΨA as well as the mixed invariants of
Ruan and Tian [67] can be viewed as special cases of the Gromov-Witten classes
as formulated by Kontsevich and Manin [35]. More precisely, let Σ be a compact
oriented Riemann surface of genus g and denote by J (Σ) the space of complex
structures on Σ which are compatible with the given orientation. A (k + 1)-tuple
(j, z1 , . . . , zk ) ⊂ J (Σ) × Σk is called stable if the only diffeomorphism φ ∈ Diff(Σ)
which satisfies φ∗ j = j and φ(zi ) = zi for i = 1, . . . , k is the identity. Let Cg,k ⊂
104
J (Σ) × Σk denote the space of such stable (k + 1)-tuples and consider the quotient
space
Cg,k
J (Σ) × Σk
Mg,k =
⊂
.
Diff(Σ)
Diff(Σ)
This is a manifold of dimension dim Mg,k = 6g − 6 + 2k. Now consider the space
Cg,k (M, A) of all (k + 2)-tuples (u, j, z1 , . . . , zk ) where (j, z1 , . . . , zk ) ∈ Cg,k and
u : Σ → M is a simple (j, J)-holomorphic curve representing the class A. The
group Diff(Σ) acts on this space by
φ∗ (u, j, z1 , . . . , zk ) = (u ◦ φ, φ∗ j, φ−1 (z1 ), . . . , φ−1 (zk ))
and, for a generic J ∈ J (M, ω), the quotient space
Mg,k (M, A) =
Map(Σ, M ) × J (Σ) × Σk
Cg,k (M, A)
⊂
Diff(Σ)
Diff(Σ)
is a manifold of dimension dim Mg,k (M, A) = (n − 3)(2 − 2g) + 2c1 (A) + 2k. This
gives rise to a linear map
GWA,g,k : H ∗ (M k ) → H ∗ (Mg,k )
defined, heuristically, by
GWA,g,k (a1 , . . . , ak ) = PD(π∗ PD(e∗1 a1 ∧ · · · ∧ e∗k ak ))
for ai ∈ H ∗ (M ) where ei : Mg,k (M, A) → M is the obvious evaluation map and
π : Mg,k (M, A) → Mg,k is the obvious projection. We shall not deal here with
the rigorous definition of these invariants for higher genus which involves suitable
compactifications of all the relevant moduli spaces. (A rigorous treatment of these
higher genus invariants will appear in the forthcoming paper [70] of Ruan–Tian.)
Note that
deg GWA,g,k (a1 , . . . , ak ) = n(2 − 2g) + 2c1 (A) −
k
X
deg ai .
i=1
The invariant Ψ corresponds to the case where this degree is zero and is given
by evaluating the cohomology class GWA,g,k (a1 , . . . , ak ) at a point. The invariant Φ on the other hand corresponds to the case where this degree agrees with
the dimension of the space Mg,k and is given by evaluating the cohomology class
GWA,g,k (a1 , . . . , ak ) on the fundamental cycle [Mg,k ]. This is what Kontsevich
and Manin call a codimension-0 class. The mixed invariants of Ruan and Tian [69]
in the case g = 0 and k ≥ 3 are given by evaluating a class GWA,0,k (a1 , . . . , ak )
of degree 2d over the 2d-dimensional cycle K(w1 , . . . , wk−d−3 ) ⊂ M0,k of all those
[j, z1 , . . . , zk ] ∈ M0,k with zi = wi for i ≤ k − d − 3.
2
Example 7.4.4 As an illustration of the difference between Φ and Ψ consider
conics in the complex projective plane. As we remarked in Section 7.3, the fact
that there is a unique conic through 5 points implies that
Φ2L (pt, pt, pt, pt, pt) = 1.
105
However, when A = 2L and all the αi are points, the dimensional condition (7.3)
is satisfied when p = 4 not p = 5. Thus
Ψ2L,p (pt, . . . , pt) = 0
unless p = 4. It is easy to check that there is exactly one map of degree 2 from
CP 1 → CP 2 such that
[1 : 0] 7→ [1 : 0 : 0],
[0 : 1] 7→ [0 : 1 : 0],
[1 : 1] 7→ [0 : 0 : 1],
[1 : x] 7→ [1 : 1 : 1],
namely [s : t] 7→ [xs(s − t) : t(s − t) : (1 − x)st]. Thus
Ψ2L (pt, pt, pt, pt) = 1.
To understand this, note that the pair (CP 1 , z) may be considered as a marked
sphere. When p > 3, these markings have moduli (i.e. they are not all equivalent
under the reparametrization group G). For example, if p = 4, the cross-ratio is
the unique invariant of z modulo the action of G. Thus, the family of all conics
which go through 4 fixed (and generic) points q1 , . . . , q4 in CP 2 has real dimension
2. The above calculation shows that for each generic marking z = {z1 , . . . , z4 },
only one conic in this family may be parametrized so that each zi is taken to the
corresponding qi .
2
In Section 9.1 we will show how to define ΨA in the weakly monotone case even
for classes A with c1 (A) = 0. Note finally that in [66] Ruan defines invariants which
count numbers of J-holomorphic tori, and uses them to obtain some information
about the symplectic topology of elliptic surfaces.
106
Chapter 8
Quantum Cohomology
Much of the recent interest in holomorphic spheres and Gromov-Witten invariants
has arisen because they may be used to define a new multiplication on the cohomology ring of a compact symplectic manifold. In the remaining chapters of this
book, we shall explain this construction, give an outline of some of the proofs,
and point out some of the relations to other subjects such as algebraic geometry,
integrable Hamiltonian systems, and Floer homology. For more details, see for example Vafa [82], Aspinwall and Morrison [2], Ruan and Tian [67] and Givental and
Kim [24].
We will begin in this chapter by assuming that M is monotone, i.e. that the
cohomology class of ω is a positive multiple of c1 . In this context, quantum cohomology can be set up with coefficients in a polynomial ring. In the first two sections,
we define the deformed cup product and explain why it should be associative. (The
proof of associativity is given in Appendix A.) We then describe the Givental–Kim
calculation of the quantum cohomology of flag manifolds and its relation to the
Toda lattice. The chapter ends with a discussion of the Gromov-Witten potential
and Dubrovin connection.
8.1
Witten’s deformed cohomology ring
Triple intersections
We shall begin by reviewing the ordinary cup product on singular cohomology.
To avoid difficulties with torsion we shall consider integral deRham cohomology
∗
H ∗ (M ) = HDR
(M, Z), i.e. deRham cohomology classes which take integral values
over all cycles. Thus we can identify H k (M ) with Hom(Hk (M, Z), Z).1 Likewise,
we denote by H∗ (M ) the free part of H∗ (M, Z) or, more precisely, the quotient of
H∗ (M, Z) by its torsion subgroup. Denote by
Z
a(β) =
a
β
1 Of
H k (M,
course, every cohomology class a ∈
Z) determines a homomorphism φa :
Hk (M, Z) → Z but this homomorphism may be zero for a nonzero cohomology class. The
torsion in H ∗ (M, Z) is, by definition, the kernel of the homomorphism a 7→ φa . In our notation
H k (M ) is isomorphic to the quotient of H k (M, Z) divided by the torsion subgroup.
107
108
CHAPTER 8. QUANTUM COHOMOLOGY
the pairing of a ∈ H k (M ) with β ∈ Hk (M ) and by α · β the intersection pairing of
two homology classes α ∈ H2n−k (M ) and β ∈ Hk (M ) of complementary dimension.
This pairing determines a homomorphism H2n−k (M ) → H k (M ) which assigns to
α ∈ H2n−k (M ) the cohomology class a = PD(α) ∈ H k (M ) defined by
Z
a=α·β
β
for β ∈ Hk (M ). Poincaré duality asserts that the map PD : H2n−k (M ) → H k (M ) is
an isomorphism. We shall denote its inverse also by PD. Now fix two cohomology
classes a ∈ H k (M ) and b ∈ H ` (M ) and denote by α ∈ H2n−k (M ) and β ∈
H2n−` (M ) their respective Poincaré duals. Then the cup product a ∪ b ∈ H k+` (M )
is defined by the triple intersection
Z
a∪b=α·β·γ
γ
for γ ∈ Hk+` (M ). Note that two cycles in general position representing α and β
intersect in a pseudo-cycle of codimension k + ` and so the triple intersection is well
defined. If a, b ∈ H ∗ (M ) are of complementary dimension then we shall use the
notation
Z
ha, bi = a · b =
a ∪ b.
M
In view of the above triple intersection formula this expression agrees with the
intersection number α · β of the corresponding Poincaré duals α = PD(a) and
β = PD(b).
P
P
Later on, we will consider elements a ∈ i H i (M ) which are are sums a = i ai
of elements ai ∈ H i (M ) of pure degree. It will then be convenient to extend the
definition of the pairing ha, bi, setting it equal to
X X
X
h
ai ,
bj i =
hai , bj i,
i
j
i,j
where hai , bj i = 0 unless i + j = 2n.
Quantum cohomology
For now, we will suppose that (M, ω) is a 2n-dimensional monotone symplectic
manifold with minimal Chern number N ≥ 2. Then, by Section 7.4, the invariant
ΨA is defined for p = 3 and p = 4 and all homology classes A. Rescaling the form
ω, if necessary, we may assume that [ω] is an integral class with ω(π2 (M )) = Z.
As explained in Chapter 1, there are several possible choices of coefficient ring
for quantum cohomology. We will begin by defining quantum cohomology as the
tensor product with the ring of Laurent polynomials
QH ∗ (M ) = H ∗ (M ) ⊗ Z[q, q −1 ].
Here q is a variable of degree 2N and so the elements in QH ∗ (M ) of degree k can
be expressed as finite sums
X
a=
ai q i ,
ai ∈ H k−2N i (M ).
i∈Z
8.1. WITTEN’S DEFORMED COHOMOLOGY RING
109
We shall denote by QH k (M ) the set of elements of degree k. There is a natural
Poincaré duality pairing QH ∗ (M ) ⊗ QH ∗ (M ) → Z : (a, b) 7→ ha, bi defined by
X
ha, bi =
ai · bj
(8.1)
2N (i+j)=k+`−2n
P
P
for a = i ai q i ∈ QH k (M ) and b = j bj q j ∈ QH ` (M ). The condition 2N (i+j) =
k +`−2n guarantees that ai and bj are of complementary dimension. We could also
sum over all pairs (i, j) and simply define ai ·bj = 0 whenever deg(ai )+deg(bj ) 6= 2n.
Note that ha, bi = 0 unless deg(a) + deg(b) ≡ 2n(mod 2N ). Note also that the
pairing (8.1) is nondegenerate in the sense that ha, bi = 0 for all b implies a = 0.
However, it need not be positive definite. It is skew commutative in the sense that
hb, ai = (−1)deg(a) deg(b) ha, bi.
for a, b ∈ QH ∗ (M ).
Remark 8.1.1 (i) Multiplication by q gives a natural isomorphism
QH k (M ) ∼
= QH k+2N (M ).
Moreover, there is an obvious isomorphism
QH k (M ) ∼
=
M
H j (M )
j≡k(mod 2N )
and hence the quantum cohomology of M can be thought of as the universal
cover of the ordinary cohomology with a Z2N grading.
(ii) We could also define QH ∗ (M ) as the tensor product with the polynomial ring
Z[q]. In fact we shall use the notation
˜ ∗ (M ) = H ∗ (M ) ⊗ Z[q]
QH
˜ k (M ) are formal
for this alternative definition. In this case the elements of QH
polynomials
X
a=
ai q i
i≥0
where again ai ∈ H k−2N i (M ). On the first glance this may seem slightly
˜ k (M )
simpler than the above definition. However, in this case the groups QH
vanish for k ≤ 0 and become periodic only for k ≥ 2n. We may in fact think
˜ ∗ (M ) as a subset of QH ∗ (M ). The restriction of the Poincaré duality
of QH
pairing (a, b) 7→ ha, bi to this subset is given by
ha, bi = a0 · b0
and is therefore degenerate.
110
(iii) As we shall see when we discuss flag manifolds in Section 8.3, it is sometimes
possible to use the coefficient ring Z[q1 , . . . , qn ] where n is the rank of H2 (M )
and the variables qi can be thought of as (multiplicative) generators of H2 (M ).
With this choice of coefficients the quantum cohomology groups, with the ring
structure defined below, will carry more information. Roughly speaking, this
is because J-holomorphic curves in different homology classes are counted
separately, while, with our simplified definition, the information about the
homology class of the curve is suppressed.
2
Deformed cup product
The deformed cup product is a homomorphism
QH ` (M ) × QH m (M ) → QH `+m (M ).
Because the Poincaré duality pairing is nondegenerate we can define the quantum
deformed cup product a ∗ b of two classes
X
X
a=
ai q i ∈ QH ` (M ),
b=
bj q j ∈ QH m (M )
i
j
in terms of their inner product with a third class
X
c=
ck q k ∈ QH 2n−`−m (M ).
k
Recall that ai ∈ H `−2N i (M ), bj ∈ H m−2N j (M ), ck ∈ H 2n−`−m−2N k (M ) and
denote their respective Poincaré duals by αi = PD(ai ), βj = PD(bj ), and γk =
PD(ck ). Then we define a ∗ b ∈ QH `+m (M ) by the formula
XX
ha ∗ b, ci =
ΦA (αi , βj , γk )
(8.2)
i,j,k A
for c ∈ QH ∗ (M ). Here the last sum runs over all classes A ∈ H2 (M ) which satisfy
N (i + j + k) + c1 (A) = 0. This is precisely the condition needed in order for the
expression ΦA (αi , βj , γk ) to be defined. In fact the codimension of the class αi (or
rather of a pseudo-cycle representing αi ) is the degree of ai and similarly for βj and
γk . These degrees satisfy the condition
deg(ai ) + deg(bj ) + deg(ck ) = 2n + 2c1 (A).
(8.3)
This corresponds precisely to the dimension conditions (7.2) and (7.3) for the definition of the invariants ΦA and ΨA . (Recall that these agree in the case p = 3.)
The formula (8.3) shows that −2n ≤ c1 (A) ≤ 2n and hence only finitely many
values of c1 (A) occur in (8.2). Because M is monotone, we have ω(A) = λc1 (A)
and hence the classes A which contribute nontrivially to the sum have uniformly
bounded energy. Thus, by Corollary 4.4.4, only finitely many such A occur and
hence the right hand side of (8.2) is a finite sum.
111
Remark 8.1.2 As discussed in chapter 1 the formula (8.2) can also be expressed
as follows. Given ordinary cohomology classes a ∈ H ` (M ) and b ∈ H m (M ) we
define
X
a∗b=
(a ∗ b)A q c1 (A)/N ∈ QH `+m (M )
d
by evaluating (a ∗ b)A ∈ H
`+m−2c1 (A)
(M ) on a class γ ∈ H`+m−2c1 (A) (M ):
Z
(a ∗ b)A = ΦA (α, β, γ).
(8.4)
γ
Here α = PD(a) ∈ H2n−` (M ), β = PD(b) ∈ H2n−m (M ), and hence the classes α,
β, γ satisfy the dimension condition
deg(α) + deg(β) + deg(γ) = 4n − 2c1 (A).
We leave it to the reader to check that this definition of a ∗ b is equivalent to (8.2).
2
Remark 8.1.3 The formula (8.2) involves only J-holomorphic curves with Chern
number 0 ≤ c1 (A) ≤ 2n. In Section 8.5 we shall discuss a family of deformed
cup-products (x, y) 7→ x ∗a y which are parametrized by cohomology classes a ∈
H ∗ (M, C). The definition of these cup-products involves the invariants ΦA,p for all
p ≥ 3 and hence J-holomorphic curves of all possible Chern numbers. The above
construction corresponds to the case a = 0 and p = 3.
2
The constant term in the expansion (8.2) comes from counting A-curves with
A = 0. Since J is ω-tame, ω restricts to an area form on any non-constant Jholomorphic curve. Thus the curves with ω(A) = 0 are constant and Φ0 (α, β, γ) is
just the usual triple intersection index
Φ0 (α, β, γ) = α · β · γ.
It follows that the constant term in the expansion of a ∗ b is just the ordinary cup
product a ∪ b. Thus this multiplication is a deformation of the cup product, as
advertised.
Proposition 8.1.4 (i) The quantum cup product is distributive over addition and
skew-commutative in the sense that
a ∗ b = (−1)deg(a) deg(b) b ∗ a
for a, b ∈ QH ∗ (M ). It also commutes with the action of Z[q, q −1 ].
(ii) If a ∈ H 0 (M ) or a ∈ H 1 (M ) then the deformed cup-product agrees with the
ordinary cup-product, i.e.
a∗b=a∪b
for all b ∈ H ∗ (M ).
(iii) The canonical generator 1l ∈ H 0 (M ) is the unit element in quantum cohomology.
112
(iv) The scalar product is invariant with respect to the cup product in the sense
that
ha ∗ b, ci = ha, b ∗ ci
for a, b, c ∈ QH ∗ (M ).
Proof: The first statement is obvious from the definition. To prove the second
statement for a ∈ H 0 (M ) or a ∈ H 1 (M ) we must prove
ΦA (α, β, γ) = 0
whenever c1 (A) 6= 0. We shall prove in fact that no A-curve will intersect both
classes β and γ when these are in general position. The double evaluation map is
defined on the space W(A, J, 2) of dimension
dim W(A, J, 2) = 2n + 2c1 (A) − 2.
Since k = 0 or k = 1 we have
2n + 2c1 (A) = deg(a) + deg(b) + deg(c) ≤ 1 + deg(b) + deg(c)
and hence
dim W(A, J, 2) ≤ deg(b) + deg(c) − 1.
But the codimension of β × γ ⊂ M 2 is deg(b) + deg(c) and is therefore larger than
the dimension of the domain W(A, J, 2) of the double evaluation map. Hence the
cycle β × γ will not intersect the image of e : W(A, J, 2) → M 2 when in general
position. This proves that the only nontrivial contribution to the deformed cupproduct occurs when c1 (A) = 0 and this is given by
Z
Z
(a ∗ b)0 = α · β · γ =
a∪b
γ
γ
This proves (ii), and (iii) follows immediately from (ii). To prove (iv) just note
that φA (αi , βj , γk ) is skew symmetric under permutations of the three entries. In
particular,
ΦA (αi , βj , γk ) = (−1)deg(a)(deg(b)+deg(c)) ΦA (βj , γk , αi )
and hence, taking the sum over all quadruples (i, j, k, A) with N (i+j +k)+c1 (A) =
0, we obtain
ha ∗ b, ci = (−1)deg(a)(deg(b)+deg(c)) hb ∗ c, ai = ha, b ∗ ci
as claimed.
2
Remark 8.1.5 A similar argument as in the proof of the previous theorem shows
that ΦA (α1 , . . . , αp ) = 0 whenever one of the classes αj has degree deg(αj ) ≥ 2n−1.
2
113
Associativity of the deformed cup-product is far from obvious and we defer this
to Section 8.2. If this is proved then, together with Proposition 8.1.4, it shows that
the quantum cohomology ring QH ∗ (M ) is a Frobenius algebra. The axioms for
a Frobenius algebra are precisely the assertions of Proposition 8.1.4 together with
associativity and skew-commutativity of the scalar product, which in our case is
given by the Poincaré duality pairing (8.1). Such algebras have many beautiful
and striking properties which are explained in the recent papers by Dubrovin [17]
and Kontsevich and Manin [35]. In particular, they give rise to flat connections
and integrable Hamiltonian systems. We shall discuss some of these structures in
Section 8.5.
Example 8.1.6 (Complex projective space) Consider CP n with its standard
complex structure and the Fubini-Study metric with corresponding Kähler form ω.
Let L ∈ H2 (CP n ) be the standard generator, represented by the line CP 1 . The
first Chern class of CP n is given by
c1 (L) = n + 1.
Hence, for reasons of dimension, the invariant ΦmL (α, β, γ) can only be non-zero
when m = 0, 1. By Proposition 8.1.4 the case m = 0 corresponds to constant curves
and gives the usual cup-product.
The minimal Chern number in this case is N = n + 1 and so the quantum
cohomology groups are given by QH k (M ) ∼
= Z when k is even and H k (M ) = {0}
`
m
when k is odd. Let a ∈ H (M ) and b ∈ H (M ). If ` + m ≤ 2n then the quantum
cup product agrees with the ordinary cup product a ∗ b = a ∪ b. So the first
interesting case is ` + m = 2n. Consider the case where a = p ∈ H 2 (M ) is the
standard generator, defined by p(L) = 1, and b = pn ∈ H 2n (CP n ). We shall
prove that the quantum product p ∗ pn is the generator q ∈ QH 2n+2 (CP n ). In
the notationR of Remark 8.1.2 we must prove that (p ∗ pn )L = 1l ∈ H 0 (CP 1 ). This
means that pt (p ∗ pn )L = 1. In view of (8.4) this integral is indeed given by
Z
(p ∗ pn )L = ΦL ([CP n−1 ], pt, pt) = 1
pt
where [CP n−1 ] = PD(p) and pt = PD(pn ). Thus we have proved
p ∗ pn = q ∈ QH 2n+2 (M ).
The other non-zero terms ΦL (α, β, γ) correspond to relations of the form pk ∗ p` =
pk+`−n−1 q for k + ` ≥ n + 1. It follows easily that quantum multiplication is associative in this case. In explicit terms the quantum cohomology ring QH ∗ (M ) has
one generator ak ∈ QH k (M ) for every even integer k and the quantum deformation
of the cup-product is now given by
ak ∗ a` = ak+`
for all k, ` ∈ 2Z. This can be expressed in the form
QH ∗ (CP n ) =
Z[p, q, q −1 ]
.
hpn+1 = q, q −1 q = 1i
114
If we tensor with the polynomial ring Z[q] then the (non-periodic) quantum cohomology groups are given by
˜ ∗ (CP n ) =
QH
Z[p, q]
.
= qi
hpn+1
Specializing to q = 0, we recover the ordinary cohomology ring of CP n .
8.2
2
Associativity and composition rules
Our goal in this section is to give an outline of the proof of the following theorem.
This result was first proved by Ruan and Tian [67, 68] and subsequently by Liu [38].
Theorem 8.2.1 (Ruan-Tian) Let (M, ω) be a compact symplectic manifold which
is monotone with minimal Chern number N ≥ 2. Then the deformed cup product
∗ on the quantum cohomology group QH ∗ (M ) is associative.
To prove this it suffices to show that the triple product
QH j (M ) ⊗ QH k (M ) ⊗ QH ` (M ) → QH j+k+` (M )
which sends (a, b, c) to (a ∗ b) ∗ c is skew symmetric with the usual sign conventions.
Now there is an obvious such skew symmetric triple product defined as follows.
Let α ∈ H2n−j (M ), β ∈ H2n−k (M ), and γ ∈ H2n−` (M ) be the Poincaré duals of
a ∈ H j (M ), b ∈ H k (M ), c ∈ H ` (M ). Then define
a?b?c=
X
(a ? b ? c)A q c1 (A)/N ∈ QH j+k+` (M )
A
by the formula
Z
(a ? b ? c)A = ΨA (α, β, γ, δ)
(8.5)
δ
for δ ∈ Hj+k+`−2N d (M ). This is well defined because M satisfies hypotheses (H4)
and (H5) in Section 7.4, and the classes α, β, γ, δ satisfy the dimension condition
deg(α) + deg(β) + deg(γ) + deg(δ) = 6n − 2c1 (A)
(8.6)
which is equivalent to (7.3).2 The triple product (8.5) is obviously skew-symmetric
and so Theorem 8.2.1 is a consequence of the following
Proposition 8.2.2 If (M, ω) is monotone with minimal Chern number N ≥ 2
then
(a ∗ b) ∗ c = a ? b ? c
for all a, b, c ∈ QH ∗ (M ).
2 Note that, already for dimensional reasons, we cannot use Φ
A in the definition (8.5) of our
triple product, since this would give a product QH j ⊗ QH k ⊗ QH ` → QH j+k+`−2 .
8.2. ASSOCIATIVITY AND COMPOSITION RULES
115
To understand the product (a ∗ b) ∗ c we shall first examine the Poincaré dual
of the cohomology class a ∗ b ∈ QH j+k (M ). This quantum cohomology class can
be thought of as a sum
X
(a ∗ b)A q c1 (A)/N
A
where the cohomology class (a ∗ b)A ∈ H j+k−2c1 (A) (M ) is defined by (8.4). We
denote the Poincaré dual of (a ∗ b)A by
ξA = PD((a ∗ b)A ) ∈ H2n+2c1 (A)−j−k (M ).
In view of (8.4) the homology class ξA is defined by the intersection property
ξA · γ = ΦA (α, β, γ)
where α = PD(a) ∈ H2n−j (M ), β = PD(b) ∈ H2n−k (M ), and γ ∈ Hj+k−2c1 (A) (M ).
With this notation the cohomology class
X
(a ∗ b) ∗ c =
((a ∗ b) ∗ c)A q c1 (A)/N
A
with ((a ∗ b) ∗ c)A ∈ H j+k+`−2c1 (A) (M ) is given by the formula
Z
X
((a ∗ b) ∗ c)A =
ΦB (ξA−B , γ, δ),
δ
(8.7)
B
for δ ∈ Hj+k+`−2c1 (A) (M ).
Our goal is to give a more direct description of the invariant ΦB (ξA , γ, δ). For
this it is necessary to find a pseudo-cycle which represents the homology class ξA .
Such a pseudo-cycle is in fact given by an evaluation map on a suitable moduli
space of J-holomorphic curves. Roughly speaking, we may think of ξA as the union
of all A-curves which go through α and β. More precisely, if z1 , z2 , z3 are three
distinct points in CP 1 then ξA can represented by the pseudo-cycle fA : V → M
where
V = {u ∈ M(A, J) | u(z1 ) ∈ α, u(z2 ) ∈ β} ,
fA (u) = u(z3 ).
Observe that the manifold V need not be compact. However, it does carry a
fundamental cycle because its boundary has codimension at least 2. The details of
this are precisely as in Chapter 6. In fact, we may consider the evaluation map
ez = eA,J,z : M(A, J) → M 3
and choose representatives of α and β such that α × β × M is transverse to ez
and to all the evaluation maps appearing in the closure of ez (M(A, J)). Then the
manifold V is the preimage
V = e−1
z (α × β × M )
and the evaluation map fA : V → M defines a pseudo-cycle.
116
Now the invariant ΦB (ξA , γ, δ) counts the B-curves which go through ξA , γ, and
δ. But every B-curve which meets ξA intersects some A-curve through α and β, and
therefore determines a cusp-curve of type (A, B). Hence the invariant ΦB (ξA , γ, δ)
counts the number of cusp curves of type (A, B) such that the A-component of
meets α and β, and the B-component meets γ and δ. Here it is possible for A or
B to be zero. For example if A = 0 then the B-curve meets α ∩ β, γ, and δ.
To formulate the invariant ΦB (ξA , γ, δ) more precisely, and to recover the symmetry between A and B, it is convenient to define an analogue of Ψ for pairs A, B
of homology classes. Fix a point z0 ∈ CP 1 and consider the space of cusp-curves
M(A, B, J) = {(u, v) ∈ M(A, J) × M(B, J) | u(z0 ) = v(z0 )} .
We have shown in Chapter 6 that this is a manifold of dimension 2n + 2c1 (A + B)
for generic J (see Example 6.4.2). Fix a quadruple
z = (z1 , z2 , z3 , z4 )
of distinct points in CP 1 and define the evaluation map
ez = eA,B,J,z : M(A, B, J) → M 4
by the formula
ez (u, v) = (u(z1 ), u(z2 ), v(z3 ), v(z4 )).
The following proposition asserts that for a generic almost complex structure J
this map determines a pseudo-cycle. The proof is precisely the same as that of
Theorem 5.4.1 and is left to the reader.
Proposition 8.2.3 Assume (M, ω) is monotone with minimal Chern number N ≥
2 Fix A, B ∈ Γ ⊂ H2 (M ) and a quadruple z = (z1 , z2 , z3 , z4 ) of distinct points in
CP 1 . Then there exists a set Jreg = Jreg (M, ω, A, B, z) ⊂ J (M, ω) of the second
category such that the evaluation map ez = eA,B,J,z : M(A, B, J) → M 4 is a
pseudo-cycle in the sense of Section 7.1 for every J ∈ Jreg . Moreover, the bordism
class of eA,B,J,z is independent of the choice of J.
Given homology classes α, β, γ, δ ∈ H∗ (M ) which satisfy the condition (8.6)
choose a pseudo-cycle f : V → M 4 which represents the class α × β × γ × δ, and
is transverse to the evaluation map eA,B,J,z and to all the evaluation maps which
occur in the description of the boundary of the set ez (M(A, B, J)). Define the
invariant ΨA,B as the intersection number
ΨA,B (α, β; γ, δ) = eA,B,J,z · f.
By Lemma 7.1.4 and Proposition 8.2.3 this integer is independent of the choice of
the pseudo-cycle f and the almost complex structure J used to define it. In fact,
the above discussion shows that
ΨA,B (α, β; γ, δ) = ΦB (ξA , γ, δ)
where ξA = PD((a ∗ b)A ). This formula can be used to express the invariant ΨA,B
entirely in terms of the original invariant ΦA . To see this choose a basis {εi }i of the
8.2. ASSOCIATIVITY AND COMPOSITION RULES
117
homology H∗ (M ) and let {φj }j be the dual basis with respect to the intersection
pairing in the sense that
φj · εi = δij .
(Note that in the odd case care must be taken with the signs.)
Lemma 8.2.4
ΨA,B (α, β; γ, δ) =
X
ΦA (α, β, εi )ΦB (φi , γ, δ).
i
Proof: Denote ei = PD(εi ), fj = PD(φj ), a = PD(α), etc. Then hfj , ei i = δij
and we obtain
X
X
ΦA (α, β, εi )fi .
h(a ∗ b)A , ei ifi =
(a ∗ b)A =
i
i
Similarly,
(c ∗ d)B =
X
hfi , (c ∗ d)B iei =
i
X
ΦA (φi , γ, δ)ei
i
Hence
h(a ∗ b)A , (c ∗ d)B i =
X
ΦA (α, β, εi )ΦB (φi , γ, δ).
i
With ξA = PD((a ∗ b)A ) as above the left hand side is given by
h(a ∗ b)A , (c ∗ d)B i = ΦB (ξA , γ, δ) = ΨA,B (α, β; γ, δ).
2
This proves the lemma.
Now we may write the formula (8.7) for (a ∗ b) ∗ c in the form
Z
X
((a ∗ b) ∗ c)A =
ΨB,A−B (α, β; γ, δ),
δ
(8.8)
B
for δ ∈ Hj+k+m−2c1 (A) (M ). Note that the classes α, β, γ, δ satisfy the dimension
condition (8.6). Comparing the formulae (8.8) and (8.5), we see that the proof
of Proposition 8.2.2, and hence that of Theorem 8.2.1, reduces to the following
identity.
Lemma 8.2.5 Assume (M, ω) is monotone with minimal Chern number N ≥ 2
and let A ∈ H2 (M ). Then
X
ΨA (α1 , α2 , α3 , α4 ) =
ΨB,A−B (α1 , α2 ; α3 , α4 )
B
for cohomology classes αj ∈ H∗ (M ) with
P
j
deg(αj ) = 6n − 2c1 (A).
Combining Lemma 8.2.5 and Lemma 8.2.4 we obtain the following composition
rule which was stated by Ruan and Tian [67] in much more generality.
Corollary 8.2.6
ΨA (α1 , α2 , α3 , α4 ) =
XX
B
i
ΨA−B (α1 , α2 , εi )ΨB (φi , α3 , α4 ).
118
To understand why Lemma 8.2.5 might be true consider the evaluation map
ez : M(A, J) → M 4
for the quadruple z = (0, 1, ∞, z) ∈ (CP 1 )4 where the point z varies in C − {0, 1}.
The elements w ∈ M(A, J) with ez (w) ∈ f are A-curves which satisfy
w(∞) ∈ α1 ,
w(1) ∈ α2 ,
w(0) ∈ α3 ,
w(z) ∈ α4 .
(8.9)
Here we assume that the product cycle α1 × α2 × α3 × α4 is transverse to the
evaluation map ez (see Remark 7.2.1). The invariant ΨA (α1 , α2 , α3 , α4 ) counts the
number of curves which satisfy this condition. The goal is to show that when z is
sufficiently close to 0 there is a one-to-one correspondence between such A-curves
and the connected pairs of curves which are counted by ΨA−B,B (α1 , α2 ; α3 , α4 ).
Consider what happens for a sequence wν ∈ M(A, J) which satisfies (8.9) when
the corresponding point zν tends to 0. Then either the sequences wν (zν ) and wν (0)
converge to the same point which therefore must lie in α3 ∩ α4 , or they converge to
different points. In the former case the limit is an A-curve which contributes to
ΦA (α1 , α2 , α3 ∩ α4 ) = ΨA,0 (α1 , α2 ; α3 , α4 ).
In the latter case, the two nearby points 0 and z get mapped to widely separated
points, and so the derivative of wν must blow up somewhere near 0. When one
rescales at the blow-up point as in the proof of Theorem 4.3.2, one will obtain a
bubble which meets α3 and α4 and lies in some class B. Generically, in view of
Lemma 6.5.1, there will only be this one bubble. Hence the restriction of wν to
compact subsets of CP 1 − {0} will converge to a curve w∞ = u ∈ M(A − B, J)
which still takes ∞ to α1 and 1 to α2 . Thus the limiting cusp-curve contributes to
ΨA−B,B (α1 , α2 ; α3 , α4 ).
Included here is the possibility that the limit curve u is constant, which corresponds
to the term
ΦA (α1 ∩ α2 , α3 , α4 ) = Ψ0,A (α1 , α2 ; α3 , α4 ).
This argument shows that if z is sufficiently close to 0 every element w which is
counted in the computation of the invariant ΨA (α1 , α2 , α3 , α4 ) has a counterpart
which contributes to the sum
X
ΨA−B,B (α1 , α2 ; α3 , α4 ).
B
To complete the proof one must show that, conversely, every pair of intersecting
curves (u, v) ∈ M(A − B, B, J) with u(CP 1 ) intersecting α1 and α2 and v(CP 1 )
intersecting α3 and α4 determines a unique curve w ∈ M(A, J) which satisfies (8.9)
provided that z is sufficiently close to 0. A proof using Floer’s original technique
involving infinite cylindrical ends has been worked out by Liu [38]. We shall present
a more direct argument in Appendix A. Ruan and Tian take a somewhat different
approach to this problem which involves the perturbed Cauchy-Riemann equations.
Their argument is outlined in [67] and the details are carried out in [68].
8.3. FLAG MANIFOLDS
119
Remark 8.2.7 An observant reader will notice that no mention has been made
above of the effect of multiply-covered curves. It would be possible for the limiting
curve to be a cusp-curve with one or both components multiply-covered. However,
the fact that each component only contains two marked points means that such an
event occurs with high codimension and so may be ignored. This may be proved
by the methods of Chapter 6.
2
Example 8.2.8 We have already seen that in CP 2 ,
Ψ2L (pt, pt, pt, pt) = 1.
Since αi ∩ αj = ∅ here, there is no contribution from decompositions 2L = A + B
where one of the classes A or B is zero. According to Lemma 8.2.5 we should
therefore have
ΨL,L (pt, pt; pt, pt) = 1.
Here the left hand side is the number of pairs of intersecting lines L1 , L2 with the
first two points on L1 and the second two on L2 . Clearly, there is a unique such
pair. In a similar way one can check this lemma for cases such as Ψ2L (pt, pt, pt, line)
in CP 3 or ΨL (pt, line, line, line) in CP 2 . It is also easy to check that Lemma 8.2.4
holds in these cases.
2
8.3
Flag manifolds
In a beautiful recent paper [24], Givental and Kim have computed the quantum
cohomology ring of the flag manifold Fn+1 , and related it to the Toda lattice. To do
this they made certain assumptions about the properties of quantum cohomology
which so far have not been established. However, their formula has been verified by
Ciocan-Fontanine in a recent paper [10]. In [3] Astashkevich and Sadov generalize
these results (in a heuristic way) to partial flag manifolds. The extreme case of
Grassmannians G(k, n) will be discussed in the next section.
Recall that the flag manifold Fn+1 is the space of all sequences of subspaces
E1 ⊂ E2 ⊂ · · · ⊂ En
of Cn+1 with dimC Ej = j. This manifold is simply connected and carries a natural complex structure. The minimal Chern number is 2. Its cohomology ring is
generated by the first Chern classes uj ∈ H 2 (M, Z) of the canonical line bundles
Lj → Fn+1
with fiber Ej+1 /Ej for j = 0, . . . , n. Since the sum L0 ⊕ · · · ⊕ Ln is trivial, these
classes are related by the condition
u0 + · · · + un = 0.
The full cohomology ring of Fn+1 is the quotient
H ∗ (Fn+1 ) =
Z[u0 , . . . , un ]
hσ1 (u), . . . , σn+1 (u)i
120
where the σj (u) denote the elementary symmetric functions. To understand this,
observe first that the cohomology classes cj = σj (u) ∈ H 2j (Fn+1 ) are the Chern
classes of the trivial bundle L0 ⊕ · · · ⊕ Ln and so must obviously be zero. The
above formula asserts that these obvious relations are the only ones and that all
the cohomology of Fn+1 is generated by the classes uj via the cup product.
The cohomology ring can also be expressed in terms of the basis
pj = uj + · · · + un
with j = 1, . . . , n. In this basis the first Chern class of the tangent bundle T Fn+1
is given by
c1 = 2(p1 + · · · + pn ).
The classes pj can be represented by forms which are Kähler with respect
to the
P
obvious complex structure J on Fn+1 . In fact a cohomology class a = j λj pj can
be represented by a Kähler form precisely if the coefficients λj are all nonnegative
and their sum is positive. In particular, there exists a Kähler form with respect
to which the manifold Fn+1 is monotone and hence quantum cohomology is well
defined. It also follows that pj (A) > 0 for every J-effective homology class A and
every j, since J is tamed by the Kähler form in class pj .
Define the degrees dj of a homology class A in terms of the homomorphisms
pj : H2 (M, Z) → Z. In other words, a homology class A ∈ H2 (M ) is uniquely
determined by the integers
dj = pj (A)
for j = 1, . . . , n and, conversely, any such set of integers determines a homology
class A = Ad . Above we associated to a class A ∈ H2 (M ) the monomial q d where
c1 (A) = N d, N is the minimal Chern number, and deg q = 2N . Now, in order to
capture the full structure of H2 (M ), we take n auxiliary variables q1 , . . . , qn each of
degree 2N = 4, which represent the basis of H2 (Fn+1 ) dual to the basis p1 , . . . , pn
of H 2 (Fn+1 ). We then represent the class A by the the monomial
q d = q1d1 · · · qndn
where d = (d1 , . . . , dn ) ∈ Zn and dj = pj (A). Since c1 = 2(p1 + · · · + pn ), this is
consistent with our previous definitions when we identify all qj with q.
Since dj = pj (A) > 0 for every effective homology class A it follows that the
pairing
X
ha ∗ b, γi =
ΦAd (α, β, γ)q d
d
(with Ad ∈ H2 (M ) determined by pj (Ad ) = dj for d ∈ Zn ) is a polynomial of
degree ` + k − j for each a ∈ H k (Fn+1 ), b ∈ H ` (Fn+1 ), and γ ∈ Hj (Fn+1 ). Hence,
we may take the coefficient ring Λ for quantum cohomology to be the polynomial
ring Z[q1 , . . . , qn ]. In other words, we define
˜ ∗ (Fn+1 ) = H ∗ (Fn+1 ) ⊗ Λ = H ∗ (Fn+1 ) ⊗ Z[q1 , . . . , qn ].
QH
Since the ui are multiplicative generators of H ∗ (Fn+1 ), the quantum cohomology
˜ ∗ (Fn+1 ) is isomorphic to some quotient of Z[u0 , . . . , un , q0 , . . . , qn ].
ring QH
8.3. FLAG MANIFOLDS
121
In [24] Givental and Kim conjectured3 that there is a natural isomorphism
Z[u0 , . . . , un , q0 , . . . , qn ]
˜ ∗ (Fn+1 ) ∼
QH
=
I
where I ⊂ Z[u0 , . . . , un , q0 , . . . , qn ] denotes the ideal generated by the coefficients
of the characteristic polynomial of the matrix


u0 q1
0
0 ···
···
0
.. 

 −1 u1 q2
0
. 



.. 
.
..
 0 −1 u2 q3

.




.
.
.
.
.
.
.
.
.
An =  0
.
.
.
0 −1
. 


 .

..
..
..
..
 ..
.
.
.
.
0 


 .

.
.
..
.. u
 ..
qn 
n−1
0 ··· ··· ···
0
−1 un
More explicitly consider the polynomials
cj = Σj (u, q)
determined by the formula
det(An + λ) = λn+1 + c1 λn + · · · + cn λ + cn+1 .
The ideal I is generated by these functions Σj . In the case q = 0 these are the
elementary symmetric functions σj (u) = Σj (u, 0) and thus the ordinary cohomology
ring appears as expected when we specialize to q = 0. In other words the classical
Chern classes are given by the elementary symmetric functions and the polynomials
Σj can be regarded as the quantum deformations of the Chern classes. A
similar phenomenon appeared in the discussion of Grassmannians in [79].
This result has some very interesting connections with completely integrable
Hamiltonian systems which we now explain. We will see that the quantum Chern
classes Σj are the Poisson commuting integrals of the Toda lattice (cf. [56]).
Recall that the Toda lattice is a Hamiltonian differential equation for n + 1 unit
masses on the real line in the positions x0 , x1 , . . . , xn . If
uj = pj − pj+1 = yj = ẋj
is the momentum of the particle at position xj and if
qj = exj −xj−1 ,
then the Hamiltonian function of the Toda lattice is given by
H(x, y) = 12 trace(A2n ) =
1
2
n
X
j=0
yj2 −
n
X
exj −xj−1 .
j=1
3 The computation of Givental and Kim assumes an equivariant version of quantum cohomology
which so far has not been established. The argument of Ciocan-Fontanine avoids this difficulty.
122
(The reversal of the usual signs corresponds to considering forces which repel when
x0 < x1 < · · · < xn .) In [56] Moser discovered that the functions Fj (x, y) =
trace(Ajn ) for j = 1, . . . , n + 1 form a complete set of Poisson commuting integrals
for this system, where the matrix An is as defined above. As a result the ideal I
generated by the quantum Chern classes Σj is invariant under Poisson brackets.
ToPstate this more precisely, observe that the standard symplectic structure
n
ω = j=0 dxj ∧ dyj , when restricted to the set x0 + · · · + xn = 0 (zero center of
mass) and y0 + · · · + yn = 0 (zero momentum) and written in terms of the variables
q1 , . . . , qn , p1 , . . . , pn , takes the form
ω=
dqn
dq1
∧ dp1 + · · · +
∧ dpn .
q1
qn
The Poisson structure is to be understood with respect to this symplectic form.
We can give the following geometric interpretation of ω. Consider the complex
torus
TC = H 2 (Fn+1 ; C/Z) = H 2 (Fn+1 ; C)/H 2 (Fn+1 ; Z)
which is parametrized by the coordinate functions
qj (a) = e2πiha,Aj i .
The cotangent bundle of TC can be naturally identified with
T ∗ TC = H 2 (Fn+1 , C/Z) × H2 (Fn+1 , C)
where the pj : H2 (M, C) → C are to be understood as coordinate functions on the
cotangent space. (These coordinate functions have a geometric meaning in terms of
the cone of Kähler forms as explained above.) Then the form ω can be understood
as a symplectic structure on the cotangent bundle T ∗ TC . In this interpretation the
above Hamiltonian system with particles x0 , . . . , xn , when restricted to the set of
zero center of mass and zero momentum, lives on the imaginary part of T ∗ T.
This leads to a geometric interpretation of the quantum cohomology itself. The
ideal
J ⊂ C[p1 , . . . , pn , q1 , . . . , qn ]
which corresponds to I determines an algebraic variety
L ⊂ T ∗ TC .
This variety is defined as the common zero set of the polynomials in J . Hence the
quantum cohomology ring
˜ ∗ (M, C) = C[p1 , . . . , pn , q1 , . . . , qn ]
QH
J
with complex coefficients can be interpreted as the space of functions on L. The
invariance of the ideal J under Poisson brackets translates into the condition that
the variety L is Lagrangian. It should therefore have a generating function and this,
according to Givental and Kim, should have a bearing on the question of mirror
symmetry.
8.4. GRASSMANNIANS
8.4
123
Grassmannians
The quantum cohomology ring of the Grassmannian G(k, N ) has been studied by
Bertram, Daskalopoulos, and Wentworth in [6], Witten in [87], and Siebert and
Tian in [79]. In [87] Witten also relates the quantum cohomology of G(k, N ) to
the Verlinde algebra of representations of U(k). His explanation for this relation is
based on a beautiful, but heuristic, consideration involving path integrals, and so
far there is no rigorous proof.
Denote by G(k, N ) the Grassmannian of k-planes in CN . Thus a point in
G(k, N ) is a k-dimensional subspace V ⊂ CN . A unitary frame of V is a matrix
B ⊂ CN ×k such that
V = im B,
B ∗ B = 1lk×k .
Two such frames B and B 0 represent the same subspace V if there exists a unitary
matrix U ∈ U(k) such that B 0 = BU . Hence the Grassmannian can be identified
with the quotient space
G(k, N ) = P/U(k)
where P ⊂ CN ×k denotes the set of unitary k-frames. A moment’s thought shows
that the Grassmannian has real dimension
dim G(k, N ) = 2k(N − k).
It can also be interpreted as a symplectic quotient. The space CN ×k is a symplectic
manifold and the natural action of the group U(k) on this space is Hamiltonian with
moment map µ(B) = B ∗ B/2i. The Grassmannian now appears as the quotient
G(k, N ) = µ−1 (1l/2i)/U(k).
Now there are two natural complex vector bundles, E → G(k, N ) of rank k and
F → G(k, N ) of rank N − k, whose Whitney sum is naturally isomorphic to the
trivial bundle G(k, N ) × CN . The fiber of E at the point V ∈ G(k, N ) is just the
space V itself and the fiber of F is the quotient CN /V :
FV = CN /V.
EV = V,
Denote the Chern classes of the dual bundles E ∗ and F ∗ by
xj = cj (E ∗ ) ∈ H 2j (G(k, N )),
yj = cj (F ∗ ) ∈ H 2j (G(k, N )).
These classes generate the cohomology of G(k, N ). Since E ⊕ F is isomorphic to
the trivial bundle there are obvious relations
j
X
xi yj−i = 0
i=0
for j = 1, . . . , N . For j > N this equation is trivially satisfied. For j = 1, . . . , N − k
it determines the classes yj inductively as functions of x1 , . . . , xk via
yj = −x1 yj−1 − · · · − xj−1 y1 − xj ,
j = 1, . . . , N − k.
For j > N − k the classes yj vanish and this determines relations of the xj . These
are the only relations and hence the cohomology ring of the Grassmannian can be
identified with the quotient
H ∗ (G(k, N ), C) ∼
=
C[x1 , . . . , xk ]
.
hyN −k+1 , . . . , yN i
124
Moreover, the first Chern class of the tangent bundle is given by c1 (T G(k, N )) =
N x1 and so the minimal Chern number is N . The following theorem was essentially
proved by Witten [87]. Independently, a rigorous proof with all details was worked
out by Siebert and Tian [79].
Theorem 8.4.1 (Siebert-Tian, Witten) The quantum cohomology of the Grassmannian is isomorphic to the ring
QH ∗ (G(k, N )) ∼
=
C[x1 , . . . , xk , q]
.
hyN −k+1 , . . . , yN −1 , yN + (−1)N −k qi
Here xj is a generator of degree 2j and q is a generator of degree 2N . The relation
yN + (−1)N −k q = 0 can also be written in the form
xk yN −k = (−1)N −k q.
Proof: We remark first that, by an easy induction argument, the classes x1 , . . . , xk
still generate the quantum cohomology of G(k, N ). This means that every cohomology class can be expressed as a linear combination of quantum products of the
xi . To prove this one uses induction over the degree and the fact that the difference
x ∗ y − x ∪ y is a sum of terms of lower degree that x ∪ y. (See Lemma 2.1 in [79] for
details.) Now this same argument shows that the original relations in the classical
cohomology ring become relations in quantum cohomology by adding certain lower
order terms. It follows again by induction over the degree that these new relations
generate the ideal of relations in quantum cohomology. (See Theorem 2.2 in [79]
for details.)
In view of these general remarks we must compute the quantum deformations of
the defining relations in the cohomology ring. In this we follow Witten’s argument
in [87]. The quantum cup product of the classes xi and yj is a power series of the
form
X
xi ∗ yj =
(xi ∗ yj )d q d
d
where (xi ∗ yj )d ∈ H 2i+2j−2N d (G(k, N )) and (xj ∗ yj )0 = xi ∪ yj . It follows that
the quantum cup-product xi ∗ yj must agree with the ordinary cupproduct xi ∪ yj
unless i = k and j = N − k. Now the relations yj = 0 for j = N − k + 1, . . . , N − 1
only involve the products xi yj with either i < k or j < N − k and hence they
remain valid in quantum cohomology. However the relation yN = 0 involves the
product xk yN −k and the only nontrivial contribution to the quantum deformation
of this product is the term (xk ∗ yN −k )1 ∈ H 0 (G(k, N )). We claim that
(xk ∗ yN −k )1 = (−1)N −k 1l.
(8.10)
To see this we must examine the moduli space M(L, i) of holomorphic curves
u : CP 1 → G(k, N ) of degree deg(u) = c1 (L) = 1. The space of such curves has
formal dimension
dim M(L, i) = 2k(N − k) + 2N.
Every holomorphic curve u ∈ M(L, i) is of the form
u([z0 : z1 ]) = span {z0 v0 + z1 v1 , v2 , . . . , vk }
(8.11)
8.4. GRASSMANNIANS
125
where the vectors v0 , . . . , vk ∈ CN are linearly independent. Note that these maps
form indeed a space of dimension 2N (k + 1) − 2k 2 which is in accordance with the
dimension formula for M(L, i). Moreover, one can check that all these curves are
regular. We must prove that
ΦL (ξk , ηN −k , pt) = (−1)N −k
where ξk = PD(xk ) and ηN −k = PD(yN −k ). Now the Poincaré dual of the topdimensional Chern class of a vector bundle can be represented by the zero set
of a generic section. For example fix a vector w ∈ Cn and consider the section
G(k, N ) → E which assigns to every k-plane V ⊂ CN the restriction of the functional v 7→ w∗ v to V . This section is transverse to the zero section and its zero set
is the submanifold
X = {V ∈ G(k, N ) | w∗ v = 0 ∀ v ∈ V } .
This submanifold is a copy of G(k, N − 1) in G(k, N ) and represents the class ξk ∈
H 2k (G(k, N )). Now fix a vector v0 ∈ CN and consider the section G(k, N ) → F
which assigns to every V ∈ G(k, N ) the equivalence class [v0 ] ∈ CN /V = FV . The
zero set of this section is the submanifold
Y = {V ∈ G(k, N ) | v0 ∈ V } .
This submanifold with its natural orientation represents the Poincaré dual of the
top Chern class cN −k (F ) = (−1)N −k cN −k (F ∗ ). In summary,
[X] = ξk ,
[Y ] = (−1)N −k ηN −k .
Finally fix any point V0 ∈ G(k, N ) − X such that v0 ∈
/ V0 and choose a basis
v1 , . . . , vk of V0 such that the vectors v2 , . . . , vk and v0 + v1 are perpendicular to w.
Then the curve u([z0 : z1 ]) = span{z0 v0 + z1 v1 , v2 , . . . , vk } satisfies
u([1 : 1]) ∈ X,
u([0 : 1]) = V0 ,
u([1 : 0]) ∈ Y.
Moreover, the intersection number at u is 1 and it is easy to see that there is no
other curve of degree 1 whose image intersects X, Y , and V0 . Hence
ΦL ([X], [Y ], pt) = 1
and this proves the formula (8.10). It follows that in quantum cohomology the
deformed relations are yj = 0 for N − k + 1 ≤ j ≤ N − 1 and yN + (−1)N −k q = 0.
This proves the theorem.
2
More details of the proof can be found in the paper [79] by Siebert and Tian.
For example they give a proof of the fact that every holomorphic curve of degree 1
in G(k, N ) must be of the form (8.11) and that they are regular in the sense that
the associated Cauchy-Riemann operator Du is surjective.
Theorem 8.4.1 shows once again how the quantum cohomology reduces to the
classical cohomology ring if we consider q to be a complex number rather than a
variable and specialize to q = 0. More explicitly, if we define the Chern polynomials
ct (E ∗ ) =
k
X
i=1
xi t i ,
ct (F ∗ ) =
N
−k
X
j=1
yj tj
126
then the classical cohomology ring is determined by the relation
ct (E ∗ )ct (F ∗ ) = 1
whereas the quantum cohomology ring is determined by
ct (E ∗ )ct (F ∗ ) = 1 + (−1)N −k qtN
(8.12)
where q ∈ C.
Landau-Ginzburg formulation
The relations in classical cohomology can be generated by the derivatives of a
single function W0 = W0 (x1 , . . . , xk ). The same is true for the relations in quantum cohomology and the corresponding function W = W (x1 , . . . , xk ) is called the
Landau-Ginzburg potential. In our discussion of this approach we follow closely
the exposition of Witten in [87].
Consider the polynomial
k
X
ct (E ∗ ) =
xi t i
i=1
∗
2i
where xi = ci (E ) ∈ H (G(k, N )). Define polynomials yj = yj (x1 , . . . , xk ) for
j ≥ 0 by the formula
X
1
=
yj tj .
ct (E ∗ )
j≥0
Then the classical cohomology ring of the Grassmannian is described by the relations yj (x) = 0 for N − k + 1 ≤ j ≤ N . These relations imply yj (x) = 0 for j > N
and, of course, for 1 ≤ j ≤ N − k the classes yj are the Chern classes of F ∗ . In
the followingP
we shall, however, not impose the condition yj (x) = 0 and then the
power series j yj tj may no longer be a polynomial.
Consider the holomorphic functions Ur = Ur (x1 , . . . , xk ) defined by the equation
X
− log ct (E ∗ ) =
Ur (x)tr .
r≥0
Differentiating this expression with respect to xj we see that
−
X ∂Ur
tj
=
tr .
∗
ct (E )
∂xj
r≥0
Comparing coefficients we find ∂Ur /∂xj = −yr−j for 1 ≤ j ≤ k. In particular, when
r − j is negative this formula implies the vanishing of the corresponding derivative
of Ur . The most interesting case is r = N + 1. With
W0 = (−1)N +1 UN +1
we obtain
∂W0
= (−1)N yN +1−j
∂xj
8.4. GRASSMANNIANS
127
for 1 ≤ j ≤ k. Hence the defining relations for the classical cohomology of the
Grassmannian can be written in the form
dW0 = 0.
Now consider the function
W = W0 + (−1)k qx1
where q is a fixed complex number. Then the condition dW = 0 is equivalent to
yN −k+1 = 0,
...
, yN −1 = 0,
yN + (−1)N −k q = 0
and these are precisely the defining relations for the quantum
cohomology ring of
P
G(k, N ). Now the higher coefficients of the power series j yj tj will no longer vanP
ish. Comparing the coefficients up to order N in the equation ct (E ∗ )·( j≥0 yj tj ) =
1 we obtain

 

k
N
−k
X
X

xi t i  · 
yj tj  = 1 + (−1)N −k qtN
j=1
j=1
and this agrees with (8.12). The function W is called the Landau-Ginzburg
potential. It can be conveniently expressed in terms of the roots λ1 , . . . , λk of the
Chern polynomial
k
k
X
Y
ct (E ∗ ) =
xi t i =
(1 + λi t),
(8.13)
i=1
i=1
namely
W (λ1 , . . . , λk ) =
k
X
i=1
λi N +1
+ (−1)k qλi
N +1
!
.
(8.14)
Geometrically, the quantum cohomology ring of G(k, N ) can be interpreted as the
ring of polynomials in the variables x1 , . . . , xk restricted to the zero set of dW . Now
the function W : Ck → C has only finitely many critical points and the equivalence
class of any polynomial f ∈ C[x1 , . . . , xk ] with respect to the ideal
∂W
∂W
Jq = hyN −k+1 , . . . , yN −1 , yN + (−1)N −k qi =
,...,
∂x1
∂xk
is determined by the values of f at the critical points of W . This gives rise to
localization formulae such as
2
−1
(−1)k(k−1)/2 X
∂ W
I(f ) =
det
f (x)
(8.15)
k!
∂xi ∂xj
dW (x)=0
for every polynomial f of degree 2k(N − k). Here the variable xi is understood to
be of degree 2i and the functional I(f ) denotes the integral of the differential form
ωf ∈ Ω2k(N −k) (G(k, N )) associated to f under the isomorphism H ∗ (G(k, N ), C) =
C[x1 , . . . , xk ]/hdW0 i.
128
Exercise 8.4.2 Prove that
Z
ck (E ∗ )N −k = 1
G(k,N )
by considering intersection points of N − k copies of G(k, N − 1) in G(k, N ). Now
check the formula (8.15) by applying it to the polynomial
f (x) = xk N −k =
k
Y
λi N −k
i=1
∗ N −k
which represents the class ck (E )
. The general case follows from this because
H 2k(N −k) (G(k, N )) is one dimensional. Hint: First use change of variables and
the residue calculus to express the sum (8.15) as a contour integral of the form
Q
Z
f · i<j (λi − λj )2
(−1)k(k−1)/2
Q
dλ1 . . . dλk
I(f ) =
k!(2πi)k
|λj |=R
i ∂W/∂λi
where R is large. Then note that the term ∂W/∂λi = λi N + (−1)k q in the denominator can be replaced by λi N − λi N −k without changing the value of the integral.
Now use the residue calculus again to evaluate the new integral. For details see [87]
and [79].
2
Relation with the Verlinde algebra
There is a beautiful relation between the classical cohomology ring of the Grassmannian G(k, N ) and the algebra of representations of the unitary group U(k)
furnished by the Chern character. First note that the isomorphism classes of finite
dimensional representations ρ : U(k) → Aut(Vρ ) form an algebra Rk with addition given by direct sum and multiplication by tensor product. More precisely one
should form the free Z-module generated by the isomorphism classes of irreducible
representations. Now there is an algebra homomorphism
Rk → H ∗ (G(k, N )) : ρ 7→ ch(Eρ )
defined as follows. Denote by Eρ → G(k, N ) the complex vector bundle associated
to a representation ρ : U(k) → Aut(V ) via Eρ = P ×ρ V . In this formula P →
G(k, N ) denotes the principal U(k)-bundle of unitary k-frames discussed above.
The Chern character of a vector bundle E can be obtained from the Chern classes
cj (E) via the formula
X
n
iξ
iξ
,
det λ1l −
=
cj (ξ)λn−j
ch(ξ) = exp trace
2π
2π
j=0
where n = rank E and ξ ∈ u(n). Alternatively, evaluate the power series ch on
the curvature of a connection on E to obtain a differential form representing the
class ch(E). This is the contents of Chern-Weil theory and we refer to Milnor–
Stasheff [55] for details. The formulae
ch(E ⊕ F ) = ch(E) + ch(F ),
ch(E ⊗ F ) = ch(E)ch(F )
8.4. GRASSMANNIANS
129
show that the Chern character determines an algebra homomorphism. This becomes in fact an isomorphism if we replace the finite dimensional Grassmannian by
the limit of G(k, N ) as N tends to ∞. This limit is in fact the classifying space of
the unitary group U(k). If we consider only a fixed Grassmannian G(k, N ) then it
is natural to restrict to a suitable subalgebra Rk,N ⊂ Rk .
Now there is a deformed product structure on the algebra Rk,N which is similar to quantum cohomology. This deformed product can roughly be described as
follows. Given a compact oriented Riemann surface Σ of genus g we consider the
moduli space MΣ of flat U(k)-connections over Σ. This is a finite dimensional
Kähler manifold (with singularities). It can be described as a quotient
MΣ =
Aflat (Σ)
G(Σ)
of flat U(k)-connections A ∈ Ω1 (Σ, u(k)) divided by the action of the gauge group
G(Σ) = Map(Σ, U(k)). There is a natural holomorphic line bundle
L → MΣ
whose curvature is the symplectic form on MΣ . We are interested in the dimension
of the space H 0 (MΣ , L⊗` ) of holomorphic sections of the `-th power of this bundle.
More generally, given marked points z1 , . . . , zp ∈ Σ and representations ρ1 , . . . , ρp
of U(k), we can construct a representation
ρ : G(Σ) → V = V1 ⊗ · · · ⊗ Vp ,
ρ(u) = ρ1 (u(z1 )) ⊗ · · · ⊗ ρp (u(zp ))
of the gauge group and form the vector bundle
E` (ρ1 , . . . , ρp ) = L⊗` ⊗ Aflat (Σ) ×ρ V
over MΣ . Now define the numbers
N` (Σ; ρ1 , . . . , ρp ) = dim H 0 (MΣ , E` (ρ1 , . . . , ρp )).
To give a precise meaning to these numbers one must define the space of holomorphic sections in the right way. This is particularly apparent in the case Σ = S 2
since in this case the moduli space of flat connections is just a point. However,
this is a point with nontrivial isotropy subgroup and the spaces of holomorphic
sections will be nontrivial. They should in fact be defined as global holomorphic
maps A(Σ) → C on the space of all connections which are invariant under a suitable action of the gauge group G(Σ). Even if this programme has been carried out
the above formula for N` (Σ; ρ1 , . . . , ρp ) is only correct if the higher cohomology
groups vanish and should otherwise be replaced by a suitable Euler characteristic.
Another difficulty arises from the presence of a U(1) factor in U(k). As a result we
must specify a pair of integers (`, m) (called the level) to characterize the required
line bundle over MΣ rather than just taking the `-fold tensor product of the given
bundle L. This would give rise to invariants N`,m . In the following we shall discard
the choice of the level in the notation and simply write N (Σ; ρ1 , . . . , ρp ) instead of
N` (Σ; ρ1 , . . . , ρp ).
130
The numbers N (Σ; ρ1 , . . . , ρp ) satisfy the Verlinde gluing rules. One such rule
is of the form
X
N (Σg+1 ; ρ1 , . . . , ρp ) =
N (Σg ; ρ1 , . . . , ρp , ρ, ρ∗ )
ρ
where the sum is over a basis of Rk,N . Here the subalgebra Rk,N ⊂ Rk must
be chosen an accordance with the level at which the invariants N (Σ; ρ1 , . . . , ρp )
are defined. For example, in the slightly simpler context of the group SU(k), if
the invariants are defined in terms of the `-the power of the canonical line bundle
L → MΣ , then the algebra should be chosen with generators sn (Ck ) (the symmetric
powers of Ck ) up to order |n| ≤ `. According to Witten, the right level to choose in
connection with the Grassmannian G(k, N ) is ` = N − k, and the additional U(1)
factor should be treated at the level m = N . Henceforth this will be our choice for
the subalgebra Rk,N ⊂ Rk and for the corresponding line bundle in the definition
of the invariants.
A similar gluing rule takes the form
X
N (S 2 ; ρ1 , ρ2 , ρ3 , ρ4 ) =
N (S 2 ; ρ1 , ρ2 , ρ∗ )N (S 2 ; ρ, ρ3 , ρ4 ).
(8.16)
ρ
This is reminiscent of the composition rules for the Gromov-Witten invariants and
can in fact be interpreted in terms of a deformed product structure on the algebra
Rk,N . The Verlinde product structure is defined by
X
ρα ∗ ρβ =
N (S 2 ; ρα , ρβ , ρ∗γ )ργ .
(8.17)
γ
The gluing rule (8.16) asserts that this product is associative. Now in [87] Witten
conjectured that there should be an isomorphism
Rk,N → QH ∗ (G(k, N )) : ρ 7→ qch(Eρ )
which maps the Verlinde product structure onto the quantum cup-product structure
of the Grassmannian. This isomorphism should be a kind of quantum deformation
of the Chern character where the deformation involves lower order terms. It is
perhaps not unreasonable to expect that the quantum Chern character of a complex line bundle Lρ → G(k, N ) should be the quantum exponential of the first
Chern class (i.e. the exponential with multiplication replaced by the quantum cupproduct). The existence of this isomorphism is a surprising and beautiful conjecture
which should have interesting consequences. Moreover, there is a generalized version of this conjecture for Riemann surfaces of higher genus with marked points if
one considers the corresponding Gromov-Witten invariants Ψ. For more details the
reader may wish to consult [87].
8.5
The Gromov-Witten potential
In a recent paper [35] Kontsevich and Manin describe various remarkable structures
related to the associativity rule and Frobenius algebra structure of quantum cohomology. Another discussion of these developments can be found in Ruan–Tian [67].
8.5. THE GROMOV-WITTEN POTENTIAL
131
To describe these structures in their full generality requires the formalism of supermanifolds. In order to avoid these additional technicalities we shall consider in this
section only the even dimensional part of the quantum cohomology groups, with
complex coefficients, and denote it by
M
H = H ev (M, C) =
H 2i (M, C).
i
The Poincaré duality pairing with complex coefficients is the Hermitian form
Z
ha, bi =
a∪b
(8.18)
M
for a, b ∈ H. This form is complex linear in the second variable, complex anti-linear
in the first variable and, because we consider only even classes, it satisfies
ha, bi = hb, ai.
Note that the restriction of this pairing to integral classes agrees with the one
defined in Section 8.1. It need not be positive definite. The extension of the
quantum cup product to complex coefficients is given by the same formula as before,
but with q set equal to 1 ∈ C. Thus
X
ha ∗ b, ci =
ΦA (α, β, γ)
A
where α = PD(a), β = PD(b), and γ = PD(c). Here we think of ΦA as being
complex linear in all variables and denote by α complex conjugation. Observe that
if a, b, c have pure degree – that is, if they each belong to some space H i (M ; C)
– the only classes A which contribute nontrivially to this sum are those for which
deg(a) + deg(b) + deg(c) = 2n + 2c1 (A). In general, we must interpret this formula
for ha ∗ b, ci as
XX
ΦA (αi , β j , γk ),
i,j,k A
P
where P αi is the decomposition of α into terms of pure degree, and similarly for
P
βj , γ k .
With this this notation H is a complex Frobenius algebra in the sense that
ha ∗ b, ci = ha, b ∗ ci.
The restriction to even classes suffices for many applications such as flag manifolds
and Grassmannians.
Following Dubrovin [17] we can interpret the quantum cup product
H × H → H : (a, b) 7→ a ∗ b
as a connection on the tangent bundle of H given by
∇Y X(a) = dX(a)Y (a) + iX(a) ∗ Y (a)
(8.19)
for a ∈ H and two vector fields X, Y : H → H. (Here we identify all the tangent
spaces of H with H in the obvious way, so that the derivative dX(a) of the map
132
X at a becomes a linear map of H to itself.) The commutativity of the quantum
product can now be interpreted as the vanishing of the torsion, associativity can be
interpreted as vanishing of the curvature and the Frobenius condition ha ∗ b, ci =
ha, b ∗ ci means that ∇ is compatible with the Hermitian structure.
Lemma 8.5.1 (i) The connection (8.19) is torsion free, i.e.
[X, Y ] = ∇Y X − ∇X Y.
(ii) The connection (8.19) is compatible with the Hermitian structure, i.e.
dhY, Zi · X = h∇X Y, Zi + hY, ∇X Zi
(iii) The connection (8.19) is flat, i.e.
∇X ∇Y Z − ∇Y ∇X Z + ∇[X,Y ] Z = 0.
(iv) The connection (8.19) satisfies
∇X 1l = iX.
Proof: Our sign convention for the Lie bracket is
[X, Y ] = dX · Y − dY · X
where dX(a) denotes the differential of the map X : H → H and should be thought
of as a linear transformation of H which takes Y to dX · Y . The first statement is
now obvious. The second statement follows from the Frobenius condition by direct
calculation:
dhY, Zi · X
= hdY · X, Zi + hY, dZ · Xi
= h∇X Y − iY ∗ X, Zi + hY, ∇X Z − iZ ∗ Xi
= h∇X Y, Zi + hY, ∇X Zi.
To prove flatness note that
∇X ∇Y Z
= ∇X (dZ · Y + iZ ∗ Y )
= d(dZ · Y + iZ ∗ Y ) · X + i(dZ · Y + iZ ∗ Y ) ∗ X
= d2 Z(Y, X) + dZ · dY · X + iZ ∗ (dY · X)
+ i(dZ · X) ∗ Y + i(dZ · Y ) ∗ X − (Z ∗ Y ) ∗ X
=
∇dY ·X Z + d2 Z(X, Y ) − Z ∗ (Y ∗ X)
+ i(dZ · X) ∗ Y + i(dZ · Y ) ∗ X.
The fourth statement is obvious.
The above connection is of a very special form because the 1-form
A : T H → End(H)
2
133
given by
Aa (x)y = ix ∗ y
is constant and does not depend on the base point a. A general connection 1-form
A ∈ Ω1 (H, End(H)) can be interpreted as a family of products
H × H → H : (x, y) 7→ x ∗a y,
(8.20)
parametrized by the elements of H itself, via the formula
Aa (x)y = ix ∗a y
(8.21)
for a ∈ H and x, y ∈ Ta H = H. The corresponding connection, when regarded as
a differential operator C ∞ (H, H) → Ω1 (H, H) is given by
∇=d+A
or, more explicitly, by
∇Y X(a) = dX(a)Y (a) + iX(a) ∗a Y (a)
(8.22)
for two vector fields X, Y : H → H. The properties of the Dubrovin connection (8.22) are related to the products (8.20) as follows. We assume here that the
map H × H × H → H : (a, x, y) 7→ x ∗a y is holomorphic and complex bilinear in
x and y. We shall consider the case where these products determine a family of
Frobenius algebra structures, one on each tangent space Ta H = H.
Lemma 8.5.2 (i) The connection (8.22) is torsion free if and only if the products (8.20) are commutative, i.e.
x ∗a y = y ∗a x
for all a, x, y ∈ H.
(ii) Assume that the connection (8.22) is torsion free. Then it is compatible with
the Hermitian structure if and only if the products (8.20) satisfy the Frobenius
condition
h x ∗ a y, zi = hx, y ∗a zi
for all a, x, y, z ∈ H.
(iii) Assume that the connection (8.22) is Hermitian and torsion free. Then the
1-form (8.21) is closed if and only if there exists a holomorphic function
S : H → C such that
h x ∗ a y, zi = ∂ 3 Sa (x, y, z)
(8.23)
(iv) Assume that the connection (8.22) is Hermitian and torsion free. Then the
1-form (8.21) satisfies
A∧A=0
if and only if the products (8.20) are all associative.
(v) The connection (8.19) satisfies ∇X 1l = iX if and only if 1l is a unit for all the
products (8.20).
134
Proof: The first two statements are proved as in Lemma 8.5.1. To prove (iii) note
that the function
φa (x, y, z) = h x ∗ a y, zi
is symmetric in x, y, z. Moreover, a simple calculation shows that its derivative
d ψa (w, x, y, z) =
φa+tw (x, y, z)
dt t=0
is symmetric in w, x, y, z if and only if the connection 1-form A given by (8.21) is
closed. Now the symmetry of ψa is equivalent to the existence of a holomorphic
function S : H → C such that
φa (x, y, z) = ∂ 3 Sa (x, y, z).
In fact, an explicit formula for S is given by
Z
1 1
S(a) =
(1 − t)2 φta (a, a, a) dt.
2 0
2
The statements (iv) and (v) are obvious.
Remark 8.5.3 The proof of statement (iii) in the previous lemma can be formulated more explicitly in terms of a complex basis e0 , . . . , em of the cohomology H = H ev (M, P
C). Then H can be identified with Cm+1 via the isomorphism
m+1
C
→ H : x 7→ i xi ei . Define the functions Aijk : Cm+1 → C by
X
a=
xi ei .
Aijk (x) = h ei ∗ a ej , ek i,
i
These functions are holomorphic by assumption. They are symmetric under permutations of i, j, k if and only if the products (8.20) are symmetric and satisfy
the Frobenius condition of (ii) in Lemma 8.5.2. If this holds then there exists a
holomorphic function S : Cm+1 → C such that
∂3S
= Aijk
∂xi ∂xj ∂xk
if and only if the derivatives ∂` Aijk are symmetric under permutations of i, j, k, `.
An explicit formula for S is given by
Z
X
1 1
S(x) =
(1 − t)2
Aijk (tx)xi xj xk dt.
2 0
i,j,k
In [35] all the conditions of Lemma 8.5.2 are formulated in local coordinates. In
particular one can introduce a matrix
gij = gji = hei , ej i
which represents the metric and obtain
X
ei ∗ a ej =
Aìj (x)e` ,
`
a=
X
i
xi ei
135
where
Aìj (x) =
X
g `k Aijk
k
and g
ij
2
represents the inverse matrix of gij .
A holomorphic function
S:H→C
is called a potential function for the Dubrovin connection (8.22) if its third
derivatives satisfy the equation (8.23). In view of Lemma 8.5.2 such a function
exists if and only if the connection is Hermitian and torsion free and the connection
1-form is closed. Conversely, if the potential function S : H → C is given and and
the products (8.20) are defined by (8.23) then these products are automatically
commutative and satisfy the Frobenius condition. Moreover, 1l is a unit if and only
if the function S satisfies
(8.24)
∂ 3 Sa (1l, x, y) = h x, yi.
Associativity translates into the remarkable system of quadratic third order partial
differential equations
X
X
∂ 3 Sa (w, x, ei )∂ 3 Sa (fi , y, z) =
∂ 3 Sa (w, z, ei )∂ 3 Sa (fi , x, y)
(8.25)
i
i
where ei denotes a basis of H and fj denotes the dual basis with respect to the
Hermitian form (8.18), i.e.
h f j , ei i = δij .
Equation (8.25) is called the WDVV-equation (as in Witten-Dijkgraaf-VerlindeVerlinde).
Remark 8.5.4 In the notation of Remark 8.5.3 with e0 = 1l the condition (8.24)
takes the form
∂3S
= gij .
0
∂x ∂xi ∂xj
and the WDVV-equation can be written as
∂3S
X
ν,µ
∂xi ∂xj ∂xν
g νµ
∂3S
∂xµ ∂xk ∂x`
=
X
ν,µ
∂3S
∂xi ∂x` ∂xν
g νµ
∂3S
∂xµ ∂xj ∂xk
for all i, j, k, `.
2
Lemma 8.5.5 Let S : H → C be a holomorphic function which satisfies (8.24)
and let the quantum multiplication x ∗a y be defined by (8.20) for a ∈ H. Then the
following are equivalent.
(i) The products x ∗a y are associative.
(ii) The Dubrovin connection (8.22) is flat.
(iii) The potential function S satisfies the WDVV-equation (8.25).
136
Proof: The equivalence of (i) and (ii) follows from the fact that the connection 1form A, given by (8.21), is closed and, in view of Lemma 8.5.2, the condition A∧A =
0 is equivalent to associativity. Since the curvature of ∇ is the endomorphism valued
2-form
FA = dA + A ∧ A
we obtain in fact that associativity is equivalent to the condition that all the connections
∇λ = d + λA
with λ ∈ R are flat.
P
To prove the equivalence of (i) and (iii) note that y = i h f i , yiei and hence
X
hx, ei ih f i , yi
hx, yi =
i
for all x, y ∈ H. This implies
X
∂ 3 Sa (w, x, ei )∂ 3 Sa (fi , y, z)
=
i
X
h w ∗ a x, ei ih f i , y ∗a zi
i
= h w ∗ a x, y ∗a zi
= h w, x ∗a (y ∗a z)i
and therefore associativity is equivalent to the condition that the left hand side
of (8.25) is symmetric under permutations of x, y, and z. This proves the lemma.
2
If we consider the quantum deformation of the cup-product which is independent
of a then the corresponding potential function S is a cubic polynomial. By definition
of the quantum cup product this cubic polynomial is given by
S3 (a) =
1 X
ΦA (α, α, α)
3!
A
where α = PD(a). It is another remarkable observation of Witten that the function
S(a) =
X 1 X
ΦA,p (α, . . . , α)
p!
p≥3
(8.26)
A
with α = PD(a) should be a solution of the WDVV-equation (8.25).PAs before,
the class a may be a sum of terms of different degrees, i.e. a =
i ai where
deg(ai ) = di . In this case the term ΦA,p (α, . . . , α) should be interpreted as the sum
of the terms ΦA,p (αi1 , . . . , αip ) over all multi-indices (i1 , . . . , ip ). The only nonzero
terms in this sum are those where A satisfies the dimension condition
2c1 (A) =
p
X
(deg(aij ) − 2) + 6 − 2n.
(8.27)
j=1
This condition is equivalent to (7.2) and guarantees finiteness of the second sum for
fixed p (in the monotone case) but there are infinitely many possibly nonzero terms
corresponding to increasing values of p. As a result there is a nontrivial convergence
137
problem, even in the monotone case. Note also the interesting case of Calabi-Yau
manifolds with n = 3 and c1 = 0. In this case the right hand side of (8.27) is zero
for a ∈ H 2 (M ), and so the dimensional condition is satisfied for all classes A. So
in this case the sum on the right is infinite, even for fixed p, and we must multiply
the terms by a factor e−tω(A) to have any hope of convergence.
The fact that the third order term in S satisfies the WDVV-equation is precisely
the formula of Corollary 8.2.6 and is, of course, equivalent to the associativity law
proved above. The proof that the whole function (8.26), if the series converges, is a
solution of (8.25) requires a gluing argument for J-holomorphic curves which, as we
now explain, involves the mixed invariants of Ruan and Tian [67] or equivalently
the higher codimension classes of Kontsevich and Manin [35].
Notice that the third derivative of the Gromov-Witten potential (8.26) is given
by
X
X
1
ΦA,p (ξ, η, ζ, α, . . . , α)
∂ 3 Sa (x, y, z) =
(p − 3)!
A
p≥3
where α = PD(a), ξ = PD(x), η = PD(y), ζ = PD(z). If the classes a, x, y, and z
are all of pure degree then the only classes A which contribute nontrivially to the
sum are those which satisfy the dimension condition
2c1 (A) = (p − 3)(deg(a) − 2) + deg(x) + deg(y) + deg(z) − 2n.
(In general this condition has to be appropriately modified.) The corresponding
product of x ∈ H k (M ) and y ∈ H ` (M ) (with k and ` even) is given by
x ∗a y =
X
p
X
1
(x ∗a y)A,p
(p − 3)!
A
m
where (x ∗a y)A,p ∈ QH (M ) is defined by
Z
(x ∗a y)A,p = ΦA,p (ξ, η, ζ, α, . . . , α)
ζ
for ζ ∈ Hm (M ) with m = k + ` + (p − 3)(deg(a) − 2) − 2c1 (A).
Now one can show as in Section 8.2 that the WDVV-equation for the GromovWitten potential translates into the composition rule
ΨA,p (θ, ξ, η, ζ; α, . . . , α) =
p−1 X
p−4
q−3
(8.28)
q=3
X
ΦB,q (θ, ξ, εi , α, . . . , α)ΦA−B,p−q+2 (φi , η, ζ, α, . . . , α)
B,i
where ΨA,p denotes the so-called mixed invariant defined with 4 marked points
on CP 1 and arbitrary intersection points otherwise, and where εi , φi are as in
Lemma 8.2.4. The equation (8.28) is correctly stated for even homology classes. In
the general case there is a similar equation which involves signs corresponding to
permutations (see Kontsevich–Manin [35] or Ruan–Tian [67]).
The proof of (8.28) involves a gluing argument as in Section 8.2 and Appendix A.
One interprets the expression on the right in terms of intersecting pairs of Jholomorphic curves, which represent the classes B and A − B and intersect the
138
appropriate homology classes. The analytical details are no different than in the
case p = 4 of 4 intersection points. An alternative proof is given in [67].
Example 8.5.6 In the case M = CP n the cohomology ring H can be naturally
identified with Cn+1 where the point z = (z0 , . . . , zn ) corresponds to the cohomology class
a = z0 1l + z1 p + z2 p2 + · · · + zn pn .
We compute the third order term S3 of the Gromov-Witten potential S. In view
of Example 8.1.6, it is not hard to check that this term is given by


X
X
1
zi zj zk +
zi zj zk  .
2
S3 (z) = 
6
i+j+k=n
i+j+k=2n+1
Example 8.5.7 In the case M = CP 1 the system (8.25) imposes no condition at
all on the function S and the Gromov-Witten potential is given by
S(z0 , z1 ) = 12 z02 z1 + ez1 − 1 − z1 − 21 z12
(cf. Kontsevich and Manin [35]). This is equivalent to the obvious fact that
ΦA,p (pt, . . . , pt) = 1 for A = [CP 1 ] and any p while all invariants with A = k[CP 1 ],
k ≥ 2, are zero.
2
Example 8.5.8 In the case M = CP 2 the equation (8.24) takes the form
∂3S
= 1,
∂z0 ∂z0 ∂z2
∂3S
= 1,
∂z0 ∂z1 ∂z1
and all other third derivatives involving z0 vanish. Moreover, it was observed by
Kontsevich and Manin in [35] that in this case the system (8.25) is equivalent to
the single differential equation
∂3S
∂3S
∂3S
+
=
∂z2 ∂z2 ∂z2
∂z1 ∂z1 ∂z1 ∂z1 ∂z2 ∂z2
∂3S
∂z1 ∂z1 ∂z2
2
.
This corresponds to the case i = j = 1, k = ` = 2 in Remark 8.5.4. Now let
N (d) = ΦdL,3d−1 (pt, . . . , pt)
be the number of rational curves of degree d in CP 2 intersecting p = 3d − 1 generic
points. This number is well defined because the condition (8.27) is satisfied with
α = [pt], deg(a) = 4, A = dL, p = 3d−1, and n = 2. The Gromov-Witten potential
is given by4
∞
X
z 3d−1 dz1
S(z) = 21 (z0 z12 + z02 z2 ) +
N (d) 2
e
(3d − 1)!
d=1
4 This formula contains a second order term 1 z 2 which corresponds to the formula Φ (pt, pt) =
L
2 2
1 and, of course, does not affect the third derivatives.
139
and the above differential equation is equivalent to the recursive formula
X
3d − 4
3d − 4
2
N (d) =
N (k)N (`)k ` `
−k
3k − 2
3k − 1
k+`=d
for d ≥ 2 with N (1) = 1 (cf. Kontsevich and Manin [35]). Observe that in this
case the recursion formula follows directly from the composition rule (8.28) and
so it holds regardless of the convergence of S. It also uniquely determines the
numbers N (d) and the Gromov-Witten potential S. The first few values of N (d)
are N (2) = 1, N (3) = 12, N (4) = 620, N (5) = 87304, N (6) = 26312976.
2
Remark 8.5.9 (i) If we include the odd-dimensional cohomology groups in the
above discussion then the commutativity and the action of the permutation
group involves signs and this leads naturally to the notion of a supermanifold.
The signs will also be involved in the correct definition of the Gromov-Witten
potential (8.26). We shall not discuss this extension here and refer the interested reader to [17] and [35].
(ii) The definition of the Gromov-Witten potential can also be extended to manifolds which are not monotone. However this would involve exponents of
the form e−tω(A) and one has to solve a nontrivial convergence problem as
explained in the next chapter.
2
140
Chapter 9
Novikov Rings and
Calabi-Yau Manifolds
The goal of this chapter is to extend the definition of the Gromov-Witten invariants
to all weakly monotone symplectic manifolds. An application to quantum cohomology for Calabi-Yau manifolds is given in Section 9.3. Recall from Remark 5.1.4 that
(M, ω) is weakly monotone if either it is monotone, or the first Chern class vanishes
over π2 (M ), or the minimal Chern number is N ≥ n − 2. So far we have only dealt
with the monotone case with minimal Chern number N ≥ 2.
In general, if the manifold M is only weakly monotone, then the dimension
condition (8.3) on c1 (A) will no longer guarantee that the energy of the curve A is
uniformly bounded and hence the sum in (8.2) will no longer be finite. In this case
the deformed cup product can be defined by
ha ∗t b, ci =
XX
ΦA (αi , βj , γk )e−tω(A)
(9.1)
i,j,k A
for c ∈ QH ∗ (M ). As before, the sum should run over all quadruples (i, j, k, A)
which satisfy N (i + j + k) + c1 (A) = 0. This is a beautiful formula and shows how
the quantum deformed cup product is actually a deformation of the ordinary cup
product for large t. However, as is often the case with beautiful formulae, there
are several problems to overcome in order to make this rigorous. The first, and
not so serious, problem is the presence of multiply covered curves of Chern number
zero. We shall see in Section 9.1 how to circumvent this difficulty. The second,
and rather more serious, problem is the question of convergence. We must find a
uniform exponential bound on the invariants ΦA (α, β, γ) in terms of ecω(A) for some
constant c. That such a bound should exist has been conjectured by physicists in
the case of Calabi-Yau manifolds (cf. [8], [9]). This convergence problem is related
to enumerative problems in algebraic geometry. Without solving this convergence
question one can work instead with formal sums and this leads naturally to the
Novikov ring. We shall discuss this approach in Section 9.2.
141
142
9.1
CHAPTER 9. NOVIKOV RINGS AND CALABI-YAU MANIFOLDS
Multiply-covered curves
We now consider the problem of extending the definition of ΦA (α, β, γ) to the case
where A = mB is a nontrivial multiple of a class B with c1 (B) = 0. For such
classes there is first of all the problem of proving that the number of simple curves
representing A (and intersecting α, β, γ) is finite under the appropriate conditions
on the dimensions of the homology classes and on the regularity of the almost
complex structure J. Secondly, there is the question of how to take account of
the multiply-covered curves which represent the class A. Moreover, in the proof of
associativity there is the difficulty of extending the definition of ΨA (α, β, γ, δ) to
cases in which condition (JA4 ) is not satisfied. In this section, we shall outline a
way to get around these difficulties.
As we have seen in Chapter 6, when A = mB is a nontrivial multiple of a class
B with c1 (B) > 0 and m > 1, then the multiply covered B-curves form part of
the boundary of the image X(A, J) of the evaluation map and so do not contribute
to ΦA . However, if c1 (B) = 0, the moduli spaces M(A, J) and M(B, J) have the
same dimension and the statements in Theorems 5.2.1 and 5.3.1 break down. To
illustrate the problems which occur in this situation, let us consider the case when
M is a 6-dimensional manifold with c1 = 0. Then the dimension of M(A, J)/G
is zero for all A and this implies that for a generic almot complex structure J all
(unparametrized) A-curves are isolated. However, if A = kB, what we have proved
so far does not imply that there are only finitely many A-curves, since there could be
a sequence of A-curves which converge to a multiply covered B-curve u : CP 1 → M .
This will not happen if J is integrable near the curve CB = u(CP 1 ) and if u is an
¯
embedding. In this case it follows from the surjectivity of the linearized ∂-operator
Du that the normal bundle of CB must have type (−1, −1), i.e. it decomposes
into a sum of holomorphic line bundles each of which has Chern number −1. This
implies that the pull-back of the normal bundle by a map of degree k has no sections
and hence, by results on complex geometry, that there are no A-curves near CB .
However, it is not clear how to extend this argument to the case of non-integrable
J. Moreover, in order to show that ΦA and ΨA are well defined, one would have
to show that such a sequence of A curves converging to a multiply covered B-curve
can only occur for a set of almost complex structures J of codimension at least 2.
Even if the above problems are solved, so that one does have a well-defined finite
number of simple A-curves, the problem remains of how to count those A-curves
which are multiple covers of some B-curve.
As noted by Ruan, one can get around both these problems by using Gromov’s
trick of considering the graphs
c = CP 1 × M,
u
b : CP 1 → M
u
b(z) = (z, u(z)),
b
of J-holomorphic curves u : CP 1 → M . These curves are J-holomorphic
with
respect to the almost complex structure
Jb = i × J
c. Moreover, if J is ω-tame (or ω-compatible) then the product structure Jb is ω
on M
btame (or ω
b -compatible) where ω
b = τ0 ×ω and τ0 is the standard symplectic form on
b
CP 1 corresponding to the Fubini-Study metric. Conversely, every J-holomorphic
9.1. MULTIPLY-COVERED CURVES
143
curve with projection φ = π1 ◦ u
b : CP 1 → CP 1 of degree 1 is the graph of a
J-holomorphic curve up to reparametrization. In [26], Gromov used this method
in a more general context to interpret graphs of solutions u : CP 1 → M of an
inhomogeneous equation of the form ∂¯J (u) = h as a Jbh -holomorphic curve, where
the complex structure Jbh depends on h.
c) and, given A ∈ H2 (M ),
Denote by A0 the homology class [CP 1 ×{pt}] ∈ H2 (M
b
b
denote A = A0 +A. Then, instead of counting A-curves in M , we can count A-curves
c
b
in M . These curves cannot be multiply covered because A is not a multiple class.
However, because we must allow for a generic perturbation of the almost complex
structure, we cannot in general work with the product structure Jb = i × J but with
an almost complex structure nearby. (In fact, it suffices to consider structures of the
form Jbh above, whose curves are graphs of the inhomogeneous equation ∂¯J (u) = h.)
For each u ∈ M(A, J) there is a 6-dimensional family of reparametrizations of
the graph u
b. This corresponds to the formal relation between the dimensions of the
moduli spaces for generic almost complex structures:
b Jb ) = dim M(A, J) + 6.
dim M(A,
c ) in
Observe further that each class α ∈ H∗ (M ) gives rise to an element in H∗ (M
1
two ways. There is the homology class of {z} × α for z ∈ CP which has the same
degree as α and also the class α
b = [CP 1 × α] of degree 2 higher.
Lemma 9.1.1 Assume that the manifold (M, ω) is weakly monotone.
(i) If the almost complex structure J ∈ J (M, ω) is semi-positive then so is the
c, ω
product structure Jb = i × J ∈ J (M
b ).
(ii) Assume that A is not a nontrivial multiple of a class B with c1 (B) = 0. Assume, moreover, that J ∈ J+ (M, ω, K) for some K > ω(A) and is regular in
the sense of Theorem 5.3.1. Then Jb is also regular and
ΦA,J (α1 , . . . , αp ) = ΦA,
b4 , . . . , α
bp )
b Jb ({z1 } × α1 , . . . , {z3 } × α3 , α
for any three distinct points z1 , z2 , z3 ∈ CP 1 .
(iii) If A = mB is a nontrivial multiple of a class B assume that either c1 (B) ≥ 3
or p ≤ 2m. Moreover, assume that J ∈ J (M, ω) is regular in the sense of
Theorem 5.4.1 and that every J-holomorphic curve has Chern number at least
2. Then Jb = i × J has the same properties and
ΨA,J (α1 , . . . , αp ) = ΨA,
b4 , . . . , α
bp )
b Jb ({z1 } × α1 , . . . , {z3 } × α3 , α
for any three distinct points z1 , z2 , z3 ∈ CP 1 .
Proof: Exercise.
2
It is not hard to see that the right hand side in these identities is well-defined for
all classes A and all p ≥ 3 whenever M is weakly monotone and the almost complex
structure Jb is sufficiently close to a product structure. In this case, a sequence
b Jb ) with u
u
bν ∈ M(A,
bν (ζj ) ∈ {zj } × αj cannot converge (modulo bubbling) to
144
a multiply covered curve. This is because the projection φν = π1 ◦ u
bν satisfies
φν (ζj ) = zj and, in the case Jb = i × J, is a holomorphic map of degree 1. These
conditions determine φν completely. In the general case, φν satisfies a slightly
perturbed inhomogeneous Cauchy-Riemann equation and the points zj will also be
perturbed because the cycles representing {zj }×αj must be put in general position.
But it is still impossible for φν to converge to a multiply covered curve. Hence the
proof of Theorem 5.4.1 shows that the codimension argument works for each class
b It follows that the invariants Φ b b and Ψ b b are always well defined and do
A.
A,J
A,J
b provided that this is sufficiently near a product
not depend on the choice of J,
structure. In view of Lemma 9.1.1 we can use these invariants to define ΦA and
ΨA for all classes A ∈ H2 (M, Z) in a weakly monotone symplectic manifold.
This is the approach taken by Ruan and Tian in [67]. It allows us to extend the
definition of quantum cohomology to all weakly monotone symplectic manifolds.
Also the proof of associativity will be essentially the same as in Section 8.2. Of
course, the problem of computing the deformed cup product will in general be
nontrivial and is a topic for future research. We give one example at the end of this
chapter.
9.2
Novikov rings
If the symplectic manifold M is not monotone then the set of J-holomorphic curves
with given Chern number will in general not have uniformly bounded energy. One
way to overcome this difficulty is to count J-holomorphic curves in a given homology class and this leads naturally to the Novikov ring Λ = Λω associated to the
homomorphism
ω : Γ → Z.
Here Γ ⊂ H2 (M ) is the image of the Hurewicz homomorphism π2 (M ) → H2 (M )
and ω is the symplectic form on M . One should think of the Novikov ring as a
completion of the group ring of Γ, which is associated to a grading of Γ induced by
ω. Thus it is more like a Laurent ring than a polynomial ring. Because we have
divided out by the torsion subgroup, Γ is isomorphic to Zm for some m and below
we shall give a description which depends on an explicit choice of this isomorphism.
But we begin our discussion in a coordinate free setting.
An element of the Novikov ring can be thought of as a Fourier series of the form
X
λ=
λA e2πiA
A∈Γ
where λA ∈ Z. Alternatively, we could allow the λA to be in any principal ideal
domain. The Novikov ring Λω consists of all formal sums λ of this form, such that,
for each c > 0, there are only finitely many nonzero coefficients λA with energy
ω(A) ≤ c. In other words, the coefficients λA are subject to the finiteness condition
# {A ∈ Γ | λA 6= 0, ω(A) ≤ c} < ∞
for every c > 0. Thus the Novikov ring is a kind of completion of the group ring of
Γ. The ring structure is given by
X
λ∗µ=
λA µB e2πi(A+B) .
A,B
9.2. NOVIKOV RINGS
145
Thus
(λ ∗ µ)A =
X
λA−B µB .
B
It is a simple matter to check that the finiteness condition is preserved under this
multiplication.
The Novikov ring carries a natural grading given by the first Chern class via
deg(e2πiA ) = 2c1 (A).
If our symplectic manifold is monotone, this grading agrees up to a positive factor
with the energy level ω(A) used in the finiteness condition. The monotonicity condition also implies that the homomorphism ω : Γ → Z has an (m−1)-dimensional kernel (if Γ ∼
= Zm ). If the monotonicity condition is dropped then ω will in general not
be an integral class. However, it does determine a homomorphism ω : H2 (M ) → R,
which may be injective. This case may occur for example in Calabi-Yau manifolds
where the first Chern class is zero. Note that in the case c1 = 0 the Novikov ring
is not graded.
An integral basis A1 , . . . , Am of Γ determines an explicit isomorphism Zm → Γ
which assigns to every integer vector d ∈ Zm the homology class
Ad = d1 A1 + · · · + dm Am .
Correspondingly, the vector d is assigned the energy level ω(Ad ). We may in this
case introduce the auxiliary variables qj = e2πiAj which we think of as multiplicative
representatives of the classes Aj With this formal notation the elements of the
Novikov ring are formal sums
X
dm
λ=
λd q d ,
q d = q1d1 · · · qm
.
d
The finiteness condition now becomes
# {d ∈ Zm | λd 6= 0, ω(Ad ) ≤ c} < ∞
and the grading is given by deg q d = 2c1 (Ad ). Note that the exponent d may have
negative components.
Now consider the special case where Γ = H2 (M ). This occurs, for example, when M is simply connected. In this case the map Zm → H2 (M ) : d 7→
Ad is an isomorphism and the components p1 , . . . , pm of the inverse isomorphism
p : H2 (M ) → Zm can be interpreted as cohomology classes in H 2 (M ). They
form a basis of H 2 (M ) which is dual to the basis A1 , . . . , Am of H2 (M ) since
pi (Aj ) = δij . Moreover, in this case we may think of the auxiliary variables qj which
were introduced above as coordinates on the cohomology groups H 2 (M, C/Z) =
H 2 (M, C)/H 2 (M, Z) given by
2πi
qj (a) = e
R
Aj
a
for a ∈ H 2 (M, C). This would give a rigorous meaning to the formal expression
qj = e2πiAj . Each coordinate function qj : H 2 (M, C/Z) → C∗ is a homomorphism
with respect to the multiplicative structure of C∗ . We took essentially this approach
when discussing flag manifolds, except that we considered only positive powers of
the generators qi .
146
Remark 9.2.1 (i) If c1 6= 0 we denote by Λk ⊂ Λ the subset of all elements of
degree k. Then Λ0 is a ring, but, in general, multiplication changes the degree
via the formula
deg(λ ∗ µ) = deg(λ) + deg(µ).
In other words, Λk is a module over Λ0 . Moreover, multiplication by any
element of degree k provides a bijection Λ0 → Λk . Note that Λk 6= ∅ if and
only if k is an integer multiple of 2N where N is the minimal Chern number
of M (defined by hc1 , π2 (M )i = N Z).
(ii) If Γ = Z then Λ is the ring of Laurent series with integer coefficients. This is
a principal ideal domain and if the coefficients are taken in a field then Λ is
a field. These observations remain valid when the homomorphism ω : Γ → R
is injective. (See for example [32].) In the case π2 (M ) = Z it is interesting
to note the difference between c1 = 0 and c1 = [ω]. In both cases Λω is the
ring of Laurent series but if c1 = 0 then this ring is not graded and in general
we cannot exclude the possibility of infinitely many nonzero coefficients. This
case appears for example when M is a quintic hypersurface in CP 4 (see below).
(iii) Novikov first introduced a ring of the form Λω in the context of his Morse
theory for closed 1-forms (cf. [58]). In that case Γ is replaced by the fundamental group and the homomorphism π1 (M ) → R is induced by the closed
1-form.
(iv) As we shall see in Chapter 10 below, the Novikov ring Λω does arise in the
context of Floer homology from a closed 1-form defined on the free loop space
of M . This was used by Hofer and Salamon in [32] to prove the Arnold
conjecture in the weakly monotone case, and is an indication of the close
connection between quantum cohomology and Floer homology.
2
Quantum cohomology with Novikov rings
In the monotone case this can be defined much as before. Again, we define
QH ∗ (M ) = H ∗ (M ) ⊗ Λ.
where Λ is now is the Novikov ring Λω . It is graded by
QH k (M ) =
2n
M
Hj (M ) ⊗ Λk−j .
j=0
Here we have dropped the notation ω in Λ and denote by Λk the elements of the
Novikov ring of degree k. Thus a class a ∈ QH k (M ) may be written in the form
X
a=
aA e2πiA ,
aA ∈ H k−2c1 (A) (M ).
(9.2)
A
The finiteness condition of the Novikov ring now becomes
# {A ∈ Γ | aA 6= 0, ω(A) ≤ c} < ∞
9.2. NOVIKOV RINGS
147
for every c ≥ 0. The Novikov ring Λω acts on QH ∗ (M ) in the obvious way by
λ∗a=
XX
A
λA−B aB e2πiA .
B
So QH ∗ (M ) is a module over Λω . If the first Chern class vanishes, then QH k (M )
is a module over Λω for every k. Note that in this case the quantum cohomology
groups are graded by the integers and we have QH k (M ) = H k (M )⊗Λω . In general
the group QH k (M ) is only a module over the subring Λ0 ⊂ Λω of all elements of
degree zero. In the monotone case, this is just the group ring over the kernel of
ω : Γ → Z. The dimension of QH k (M ) as a module over Λ0 is
dimΛ0 QH k (M ) =
X
bj
j≡k (mod 2N )
where bj = dim H j (M ) is the j-th Betti number of M .
Remark 9.2.2 If the first Chern class c1 does not vanish, then the quantum cohomology is periodic with period 2N where N is the minimal Chern number. In
other words there is an isomorphism
QH k (M ) ∼
= QH k+2N (M )
given by multiplication with a monomial e2πiA where A has Chern class c1 (A) = N .
This isomorphism depends on the choice of the monomial and so is not unique
(unless H2 (M ) = Z).
2
Deformed cup product
As before, the deformed cup product is the homomorphism
QH k (M ) × QH ` (M ) → QH k+` (M )
defined by
a∗b=
X
(a ∗ b)A e2πiA
A
for a ∈ H k (M ) and b ∈ H ` (M ), where (a ∗ b)A ∈ H k+`−2c1 (A) (M ) is given by
Z
(a ∗ b)A = ΦA (α, β, γ)
(9.3)
γ
for γ ∈ Hk+`−2c1 (A) (M ) where α = PD(a) ∈ H2n−k (M ),β = PD(b) ∈ H2n−` (M ).
Here the invariant ΦA is defined whenever A is a multiple class by the right hand
side of the identity in Lemma 9.1.1 (ii). Apart from this, the previous discussion
carries over without essential change. For example, formula (8.7) becomes
Z
X
((a ∗ b) ∗ c)A =
ΦA−B (ξB , γ, δ)
δ
B
148
where ξB = PD((a ∗ b)B ) and γ = PD(c). Thus, in the weakly monotone case,
quantum cohomology with coefficients in the Novikov ring Λω is well-defined and
has an associative multiplication. Moreover, as before, the constant term in the
expansion (9.3) is just the usual triple intersection index and so corresponds to the
ordinary cup product a ∪ b.
Theorem 9.2.3 Assume that (M, ω) is weakly monotone. Then the deformed cup
product on QH ∗ (M ) is associative, distributive, and skew-commutative. Moreover,
if a ∈ H 0 (M ) or a ∈ H 1 (M ) then the deformed cup-product a ∗ b agrees with the
ordinary cup-product a ∪ b. In particular, the canonical generator 1l ∈ H 0 (M ) is
the unit element.
9.3
Calabi-Yau manifolds
There are some interesting phenomena in the structure of the quantum cohomology
ring QH ∗ (M ) for general weakly monotone symplectic manifolds which do not
appear in the monotone case. For example the homomorphism ω : Γ → R need no
longer be integral, and this has its consequences for the Novikov ring Λω . The first
interesting case for this phenomenon is that of symplectic manifolds with vanishing
first Chern class.
A class of symplectic manifolds which satisfy this condition are the Calabi-Yau
manifolds. These are complex Kähler 3-folds M (6 real dimensions) with c1 = 0.
By a theorem of Yau, such manifolds admit Kähler metrics with vanishing Ricci
tensor. Calabi-Yau manifolds have found considerable interest in the recent physics
literature (see for example [8] and [9]), and the mirror symmetry conjecture which
arises in this context has greatly intrigued mathematicians. All this formed part of
the motivation for the development of quantum cohomology.
A particular example of a Calabi-Yau manifold is a quintic hypersurface in
CP 4 . Another such example is a product M = T2 × X where X ⊂ CP 3 is a quartic
hypersurface (the famous K3-surface). Recall that in a Calabi-Yau manifold the
dimension of the moduli space M(A, J)/G is zero for every class A and hence, for a
generic almost complex structure J, every simple J-holomorphic curve is isolated.
It has been conjectured by physicists that, in the case where M is a quintic in
CP 4 , this should continue to hold for generic complex structures. However, for
general Calabi-Yau manifolds with b2 > 1 it is necessary to consider non-integrable
almost complex structures in order prove the existence of only finitely many jholomorphic curves in each given homology class. An explicit counterexample is
given by P.M.H. Wilson in [84].
Convergence
An alternative (conjectural) definition of quantum cohomology can be given with
QH k (M ) = H k (M, C) in which the deformed cup product depends on a complex
parameter t with sufficiently large real part. For two classes A ∈ H k (M, C) and
b ∈ H ` (M, C) the product a ∗t b ∈ H k+` (M, C) should be defined by
Z
X
a ∗t b =
ΦA (α, β, γ)e−tω(A)
(9.4)
γ
A
9.3. CALABI-YAU MANIFOLDS
149
for γ ∈ Hk+` (M, C). It was conjectured by physicists that the series should converge
for Calabi-Yau manifolds. If this is the case, then the quantum deformation of the
cup product is an actual deformation of the ordinary cup product with deformation
parameter t, and the ordinary cup product appears as the limit when t → ∞.
Note also that, in the case of Calabi-Yau manifolds, the only interesting cohomology group, from the point of view of quantum cohomology, is H 1,1 (M ) and,
because H 2,2 (M ) ∼
= H 1,1 (M ) we can think of quantum cohomology as a product structure on H 1,1 . In fact all other groups, apart from H 0,0 (M ) = Z and
H 3,0 (M ) = Z are either zero or, as in the case of H 1,2 (M ), will only have trivial
quantum-cup-products. Now the deformation ring of M is a ring structure on H 1,2
which gives rise to a differential equation similar to the WDVV-equation above. The
mirror symmetry conjecture states that associated to each Calabi-Yau manifold M there is a mirror manifold M ∗ such that the rings H 1,1 (M ∗ ) and H 1,2 (M )
are naturally isomorphic.
Quintic hypersurfaces in CP 4
Consider the hypersurface of degree k in CP 4


4


X
Zk = [z0 : · · · : z4 ] ∈ CP 4 |
zj k = 0 .


j=0
This manifold is simply connected and has Betti numbers
b2 = b4 = 1,
b3 = k 4 − 5k 3 + 10k 2 − 10k + 4.
In particular the identity b2 = b4 = 1 follows from the Lefschetz theorem on hyperplane sections. It follows that π2 (Zk ) = Z and the symplectic form ω does not
vanish over π2 (Z). Moreover the first Chern class of Zk is given by
c1 = (5 − k)ι∗ h
where h ∈ H 2 (CP 4 , Z) is the standard generator of the cohomology of CP 4 and
ι : Zk → CP 4 is the natural embedding of Zk as a hypersurface in CP 4 .
Now let A ∈ π2 (Zk ) be the generator of the homotopy group with ω(A) > 0.
An explicit representative of A is given, for example, by the holomorphic curve
[z0 : z1 ] 7→ [z0 : z1 : −z0 : −z1 : 0] when k is odd. Evaluating the first Chern class
on this generator gives
c1 (A) = 5 − k.
So for k ≤ 4 the manifold Zk is monotone. For k ≤ 3 the minimal Chern number
is at least 2 and so the quantum cohomology can be defined with the methods of
Section 8.1. For k = 4 the minimal Chern number is 1. In this case the quantum
cohomology can still be defined with the techniques of Section 8.1, but to prove
associativity one has to combine the methods of Section 8.2 with those of Section 9.1. For k > 5 there are no nonconstant J-holomorphic curves for a generic
almost complex structure because either ω(A) ≤ 0 or c1 (A) < 0. In the latter case,
there cannot be any J-holomorphic curves for dimensional reasons because in 6
dimensions dim M(A, J)/G = 2c1 (A) < 0.
150
Hence the most interesting case is that of the quintic hypersurface M = Z5
with Chern class zero. This is the archetypal example of a Calabi-Yau manifold. In
order to compute the quantum product, we need to calculate all non-zero invariants
ΦA (α1 , α2 , α3 ). We would like the answer to be given in terms of the numbers nd
of simple J-holomorphic curves of degree d = ω(A). (Here ω is normalized to give
an isomorphism H2 (M ) → Z.) The answer should depend only on the original
manifold M , and not on some perturbation of J.
For A 6= 0 it is easy to check that, for dimensional reasons, the invariant
ΦA (α1 , α2 , α3 ) is non-zero only when each αi has dimension 4. More precisely,
by condition (8.3), the sum of the degrees of the αi must equal 12. Since the curves
in M are all isolated a generic representative of any cycle αi with degree deg(αi ) < 4
will avoid all nonconstant J-holomorphic curves. Therefore, since b4 = 1, it suffices
to calculate
ΦA (H, H, H)
where H = [Z5 ∩ CP 3 ] ∈ H4 (Z5 ) is the class of the hyperplane section.
Observe first that, because Z5 is a hypersurface in CP 4 of degree 5, we have
Φ0 (H, H, H) = H · H · H = [CP 1 ] · Z5 = 5.
Moreover, if C is a simple curve in class A of degree d then C · H = d. Thus, if
we perturb H to general position, C intersects H in d distinct points and it follows
easily that the contribution of the curve C to ΦA (H, H, H) is exactly d3 .
Now let us consider the contribution of C to ΦmA (H, H, H). Choose a holomorphic parametrization u : CP 1 → Z5 of C, let φ : CP 1 → CP 1 be a rational map of
degree m, and consider the graph vb : CP 1 → CP 1 × Z5 of u ◦ φ:
vb(z) = (z, u(φ(z)).
This curve is not regular with respect to the complex structure i×J. It belongs to a
family of curves M(A0 + mA, i × J) of dimension 6 + 4m + 2 while the dimension in
the regular case should only be 8+2c1 (mA) = 8. (Here 4m+2 is the real dimension
of the space Ratm of parametrizations φ. One can also verify that vb is non-regular
by using Lemma 3.5.1.) Thus, in order to see how C contributes to ΦmA (H, H, H),
one should perturb the complex structure i × J to a generic element Jb and count
b
the number of J-holomorphic
(A0 + mA)-curves near C. By assumption, there
are only finitely many simple J-holomorphic mA-curves which are all regular, and
these are therefore separated from C. Hence under the perturbation j × J → Jb
their graphs move slightly but remain separated from C. Similarly, if A = m0 A0 ,
the perturbation of any m0 -fold cover of an A0 -curve is separated from C. Thus one
b
can isolate the contribution from C to M(A0 + mA, J).
No one has yet performed this calculation. However, Aspinwall and Morrison
in [2] give a different but very natural calculation of this contribution using methods
coming from algebraic geometry. Their answer is that each m-fold cover of C should
also contribute d3 to ΦmA (H, H, H). (This means that there should be exactly
one element of M(A0 + mA, Jb ) coming from C.) Thus, their definition gives the
9.3. CALABI-YAU MANIFOLDS
151
following beautiful formula for a ∗ a when a ∈ H 2 (M ) is dual to H
Z
XZ
a∗a =
(a ∗ a)A e2πiA
H
A
=
X
H
ΦA (H, H, H)e2πiA
A
=
5+
∞
X
X
d3 (q d + q 2d + q 3d + . . . )
d=1 deg(C)=d
=
5+
∞
X
d=1
nd d3
qd
.
1 − qd
Here we use the notation e2πiA = q d where A ∈ H2 (M ) is the unique homology
class of degree d. The second sum is over all simple holomorphic curves C of degree
d and nd is the number of such curves.
It seems very likely that the calculation of a ∗ a using the perturbation Jb will
agree with this Aspinwall–Morrison formula. Observe also that, if we insert q = e−t
then the deformed cup-product is given by the formula
!
∞
−td
X
3 e
a ∗t a = 5 +
nd d
a
1 − e−td
d=1
and the convergence problem is now restated in terms of the convergence of the
series on the right hand side.
152
Chapter 10
Floer Homology
There is a completely different approach to quantum cohomology arising from
Floer’s proof of the Arnold conjecture and the resulting notion of Floer homology (c.f. [18] and [20]). In this chapter, we will actually construct a cohomology
theory, rather than a homology theory. We briefly describe the construction of
Floer cohomology and outline a proof that it is isomorphic to the (additive) quantum cohomology defined above. There is also a natural ring structure on Floer
cohomology, and we discuss the conjecture that this is isomorphic to the deformed
cup product in quantum cohomology.
Floer originally developed his homology theory in the context of monotone symplectic manifolds. This was later extended by Hofer and Salamon in [32] and Ono
in [60] to the weakly monotone case and it is this extension which we shall explain
below.
10.1
Floer’s cochain complex
Let (M, ω) be a compact weakly monotone symplectic manifold and let Ht = Ht+1 :
M → R be a smooth, 1-periodic family of Hamiltonian functions. Denote by
Xt : M → T M the Hamiltonian vector field defined by ι(Xt )ω = dHt and consider
the time dependent Hamiltonian differential equation
ẋ(t) = Xt (x(t)).
(10.1)
The solutions of this equation generate a family of symplectomorphisms ψt : M →
M via
d
ψt = Xt ◦ ψt ,
ψ0 = 0.
dt
It is not hard to see that the fixed points of the time-1-map ψ = ψ1 are in one-toone correspondence with the 1-periodic solutions of (10.1). We now explain how to
interpret the contractible 1-periodic solutions as the critical points of the symplectic
action functional on the universal cover of the space LM of contractible loops in
M.
For every contractible loop x : R/Z → M there exists a smooth map u : B → M
defined on the unit disc B = {z ∈ C | |z| ≤ 1} which satisfies u(e2πit ) = x(t). Two
153
154
CHAPTER 10. FLOER HOMOLOGY
such maps u1 and u2 are called equivalent if their sum u1 #(−u2 ) is homologous
to zero (in the space H2 (M ) of integral homology divided by torsion). We use the
notation
[x, u1 ] ∼ [x, u2 ]
˜ the space of equivalence classes. The elfor equivalent pairs and denote by LM
g will also be denoted by x̃. The space LM
g is the unique covering
ements of LM
space of LM whose group of deck transformations is the image Γ ⊂ H2 (M ) of the
Hurewicz homomorphism π2 (M ) → H2 (M ). We denote by
g → LM
g : (A, x̃) 7→ A#x̃
Γ × LM
g.
the obvious action of Γ on LM
g → R is defined by
The symplectic action functional aH : LM
Z
Z 1
∗
aH ([x, u]) =
u ω+
Ht (x(t)) dt
B
0
and satisfies
aH (A#x̃) = aH (x̃) + ω(A).
(10.2)
This function can therefore be interpreted as a closed 1-form on the loop space LM
g , which is exactly the situation
rather than a function on the covering space LM
considered by Novikov: cf Remark 9.2.1.
It is not hard to check that the critical points of aH are precisely the equivalence
classes [x, u] where x(t) = x(t + 1) is a contractible periodic solution of (10.1). We
e
g the set of critical points and by P(H) ⊂ LM the
shall denote by P(H)
⊂ LM
corresponding set of periodic solutions. Floer homology is essentially an infinite
dimensional version of Morse-Novikov theory for the symplectic action functional.
Consider the (upwards) gradient flow lines of aH with repect to an L2 -metric on
LM which is induced by an almost complex structure on M . These are solutions
u : R2 → M of the partial differential equation
∂u
∂u
+ J(u)
− ∇Ht (u) = 0
∂s
∂t
(10.3)
with periodicity condition u(s, t + 1) = u(s, t) and limit condition
lim u(s, t) = x± (t)
s→±∞
(10.4)
where x± ∈ P(H). We denote by M(x̃− , x̃+ ) = M(x̃− , x̃+ , H, J) the space of all
solutions of (10.3) and (10.4) with x̃− #u = x̃+ . The elements of this space have
finite energy
Z1 Z∞
E(u) =
1
2
0 −∞
2 2 !
∂u ∂u
+
ds dt = aH (x̃+ ) − aH (x̃− ).
∂s ∂t − Xt (u)
Moreover, for a generic Hamiltonian function H : M × R/Z → R, the space
M(x̃− , x̃+ ) is a finite dimensional manifold of dimension
dim M(x̃− , x̃+ ) = µ(x̃+ ) − µ(x̃− ).
10.1. FLOER’S COCHAIN COMPLEX
155
e
Here the function µ : P(H)
→ Z is a version of the Maslov index due to Conley
and Zehnder. The integer µ([x, u]) is defined by trivializing the tangent bundle
over the disc u(B) and considering the path of symplectic matrices generated by
the linearized Hamiltonian flow along x(t). We refer to [75] and [15] for more details.
Here we only point out that
µ(A#x̃) = µ(x̃) + 2c1 (A)
(10.5)
e
for x̃ ∈ P(H)
and A ∈ Γ. Moreover, the index can be normalized so that
µ(x̃) = indH (x)
(10.6)
whenever Ht ≡ H is a C 2 -small Morse function and x̃ = [x, u], where x(t) ≡ x is a
critical point of H and u(z) = x is the constant disc. The right hand side in (10.6)
is then to be understood as the Morse index.
In the case µ(ỹ) − µ(x̃) = 1, the space M(x̃, ỹ) is a 1-dimensional manifold, and
so each point in the quotient M(x̃, ỹ)/R (with R acting by time shift) is isolated.
Moreover, for a generic Hamiltonan H we have the following finiteness result.
Proposition 10.1.1 For a generic almost complex structure J and generic Hamiltonian H we have
X
# {M(x̃, A#ỹ)/R} < ∞
ω(A)≤c
c1 (A)=0
e
for all x̃, ỹ ∈ P(H)
with µ(ỹ) − µ(x̃) = 1 and every constant c.
To prove this, one has to show that the relevant moduli spaces are compact.
This will be the case if no bubbling occurs. The key observation is that, for a
generic almost complex structure J, the set of points lying on a J-holomorphic
sphere of Chern number 0 forms a set in M of codimension 4 and so, for a generic
H, no such sphere will intersect an isolated connecting orbit. Thus, it follows
from Gromov’s compactness theorem that they cannot bubble off. Moreover, Jholomorphic spheres of negative Chern number do not exist by weak monotonicity.
J-holomorphic spheres of Chern number at least 1 connot bubble off because otherwise in the limit there would be a connecting orbit with negative index difference
but such orbits do not exist generically. This is the essence of the proof of Proposition 10.1.1. Details are carried out in [32].
Whenever µ(ỹ) − µ(x̃) = 1 we denote
n(x̃, ỹ) = # {M(x̃, ỹ)/R} ,
where the connecting orbits are to be counted with appropriate signs determined
by a system of coherent orientations of the moduli spaces of connecting orbits as
in [21]. These numbers determine a cochain complex as follows. Define
CF k = CF k (H)
as the set of formal sums
ξ=
X
µ(x̃)=k
ξx̃ hx̃i
156
which satisfy the finiteness condition
n
o
e
# x̃ ∈ P(H)
| ξx̃ 6= 0, aH (x̃) ≤ c < ∞.
e
Here the generators hx̃i of the Floer complex run over the set P(H)
of critical points
of the action functional, and the coefficients ξx̃ may be taken in Z or in Q. This
complex CF ∗ is a module over the Novikov ring Λ = Λω defined above, with action
given by the formula
XX
λ∗ξ =
λA ξ(−A)#x̃ hx̃i.
x̃
A
Note that this action will change the degree, unless λA is nonzero only when c1 (A) =
0. Further, the dimension of CF ∗ (H) as a module over Λω is precisely the number
#P(H) of contractible periodic solutions of the Hamiltonian system (10.1).
The above numbers n(x̃, ỹ) determine a coboundary operator
δ : CF ∗ (H) → CF ∗ (H)
defined by
δ x̃ =
X
n(x̃, ỹ)hỹi.
µ(ỹ)=k+1
Proposition 10.1.1 and the formula (10.2) guarantee the finiteness condition required for
δ x̃ ∈ CF k+1 (H).
Floer’s proof that the square of this operator is zero carries over to the weakly
monotone case. Here the key observation is that 1-parameter families of connecting
orbits with index difference 2 will still avoid the J-holomorphic spheres of Chern
number 0 because they form a 3-dimensional set in M while these J-holomorphic
spheres form a set of codimension 4. Similarly, holomorphic spheres of Chern
number 1 can only bubble off if they intersect a periodic solution, and this does
not happen for a generic H because the points on these spheres form a set in M of
codimension 2 while the periodic orbits form 1 dimensional sets. For J-holomorphic
spheres with Chern number at least 2 the same argument as above applies. Hence no
bubbling occurs for connecting orbits with index difference 2 and hence such orbits
can only degenerate by splitting into a pair of orbits each with index difference 1.
As in the standard theory (cf. [20], [43], [75]) this shows that δ ◦ δ = 0. Hence the
solutions of (10.3) determine a cochain complex (CF ∗ , δ), and its homology groups
HF ∗ (M, ω, H, J) =
ker δ
im δ
are called the Floer cohomology groups of the pair (H, J). Because the coboundary map is linear over Λω it follows that the Floer cohomology groups form a module
over Λω . In [32] it is proved that the Floer cohomology groups are independent of
the almost complex structure J and the Hamiltonian H used to define them.
Theorem 10.1.2 For two pairs (H α , J α ) and (H β , J β ), which satisfy the regularity requirements for the definition of Floer cohomology, there exists a natural
isomorphism
Φβα : HF ∗ (M, ω, H α , J α ) → HF ∗ (M, ω, H β , J β ).
10.1. FLOER’S COCHAIN COMPLEX
157
If (H γ , J γ ) is another such pair then
Φγα = Φγβ ◦ Φβα ,
Φαα = id.
These isomorphisms are linear over Λω .
The theorem is proved by choosing a homotopy (Hs , Js ) from (H α , J α ) to
(H , J β ) and considering the finite energy solutions of the following time dependent
version of equation (10.3)
β
∂u
∂u
+ Js (u)
− ∇Hs,t (u) = 0.
∂s
∂t
Here u satisfies the usual periodicity condition u(s, t + 1) = u(s, t). Any such
solution will have limits
lim u(s, t) = xα (t),
s→−∞
lim u(s, t) = xβ (t)
s→+∞
where xα ∈ P(H α ) and xβ ∈ P(H β ). These solutions determine a chain map
CF ∗ (H α ) → CF ∗ (H β ) which is of degree zero and, choosing a homotopy of homotopies, one can see that the induced map on Floer cohomology is independent
of the choice of the homotopy. These arguments are again precisely the same as in
Floer’s original proof in [20] for the monotone case, and for the present case they
are carried out in [32]. (See also [75] for the case c1 = [ω] = 0.)
Now one can specialize to a time independent Hamiltonian function and prove
that the Floer cohomology groups are naturally isomorphic to the cohomology of
the underlying manifold M with coefficients in the Novikov ring Λω . But these are
precisely the quantum cohomology groups of M . (In order not to be concerned
with torsion, we take here cohomology with rational coefficients, replacing Λω by
Λω ⊗ Q.)
The following theorem was proved by Floer [20] in the monotone case and by
Hofer and Salamon [32] in the case where either c1 (A) = 0 for all A ∈ π2 (M ) or
the minimal Chern number is N ≥ n. The general case is treated with different
methods in [63].
Theorem 10.1.3 Assume that (M, ω) is weakly monotone. Then, for every regular
pair (H α , J α ), there exists an isomorphism
Φα : HF ∗ (M, ω, H α , J α ) → QH ∗ (M ),
where the cohomology groups are to be understood with rational coefficients. These
maps are natural in the sense that
Φβ ◦ Φβα = Φα
and they are linear over Λω .
Floer’s original proof for the monotone case and the proof in [32] are based on
the following idea. Choose a time independent Hamiltonian H : M → R which
is a Morse function. If H is sufficiently small in the C 2 -norm then the 1-periodic
solutions of (10.1) are precisely the critical points of H and, by (10.6), their Maslov
158
index agrees with the Morse index. Moreover, the gradient flow lines u : R → M
of H are solutions of the ordinary differential equation
u̇(s) = ∇H(u(s))
and they form special solutions of the partial differential equation (10.3), namely
those which are independent of t. These solutions determine the Morse-Witten
coboundary operator
δMW : C ∗ (H) → C ∗ (H).
This coboundary operator is defined on the same cochain complex as the Floer
coboundary δ and, by (10.6), the cochain complex has the same grading for both
theories. But the homology of the Morse-Witten coboundary operator is naturally
isomorphic to the quantum cohomology of M
QH ∗ (M ) =
ker δMW
.
im δMW
(See [86], [73], or Schwarz’s book [76].) Thus, to prove Theorem 10.1.3 one must
show that all the solutions of (10.3) and (10.4) with
µ(u) = µ(x̃+ ) − µ(x̃− ) ≤ 1
are independent of t, provided that Ht ≡ H is a Morse function which is independent of t and is rescaled by a sufficiently small factor.
To prove this result without any bound on the energy of the solution requires
one to assume that either M is monotone, or c1 (A) = 0 for all A ∈ π2 (M ), or the
minimal Chern number is N ≥ n. In the other cases of weak monotonicity (where
the minimal Chern number is N = n − 1 or N = n − 2) it has so far only been
possible to prove t-independence for the solutions of (10.3) with a given bound on
the energy, with the required smallness of H depending on this bound. To use such
a result for the proof of Theorem 10.1.3 one needs an alternative definition of Floer
cohomology. (First truncate the chain complex and then take inverse and direct
limits.) These ideas are due to Ono and in [60] he proved Theorem 10.1.3 with this
modified definition of Floer cohomology.
To prove this theorem with the original definition of the Floer groups requires
a different approach which was found by Piunikhin–Salamon–Schwarz [63]. The
idea is to consider perturbed J-holomorphic planes u : C → M such that u(re2πit )
converges to a periodic solution xα (t) of the (time dependent) Hamiltonian system
H α as r → ∞ and such that u(0) lies on the unstable manifold of a given critical
point x of the Morse function H : M → R (with respect to the upward gradient
flow). One can think of these as J-holomorphic spiked disks where the spike is the
gradient flowline from x to u(0). In the case where the index difference is zero the
moduli space of such spiked disks is 0-dimensional and hence the numbers n(x, x̃α )
of its elements can be used to construct a chain map C ∗ (H) → CF ∗ (H α ). In [63]
it is shown that this map induces an isomorphism on cohomology.
As a corollary of Theorem 10.1.3 it follows that all weakly monotone symplectic manifolds (M, ω) satisfy the Arnold conjecture. To see this recall that the
dimension of the Floer chain complex CF ∗ (H α ) as a module over the Novikov ring
is the number of contractible periodic solutions.
10.2. RING STRUCTURE
159
Corollary 10.1.4 (Arnold conjecture) [20, 32, 60] Let (M, ω) be a weakly
monotone compact symplectic manifold and ψ : M → M be a Hamiltonian symplectomorphism with only nondegenerate fixed points. Then
#Fix(ψ) ≥
2n
X
bj (M )
j=0
where bj (M ) = dim Hj (M ) denote the Betti numbers of M .
10.2
Ring structure
There is a natural ring structure on Floer cohomology which is defined using trajectories which connect three periodic orbits, in contrast to the cylindrical trajectories
used above to define the boundary operator. The domain of these trajectories is
the “pair of pants” Σ, and we begin by describing this space.
Σ is a noncompact Riemann surface of genus zero with three cylindrical ends,
provided with a conformal structure which is the standard product structure on
the cylindrical ends. To define this precisely, fix parametrizations
φ1 , φ2 : (−∞, 0) × R/Z → Σ,
φ3 : (0, ∞) × R/Z → Σ
of the cylindrical ends with disjoint images Uj = φj ((−∞, 0) × R/Z) for j = 1, 2
and U3 = φ3 ((0, ∞) × R/Z). Assume that the complement
Σ0 = Σ − U1 − U2 − U3
is diffeomorphic to a 2-sphere with three open discs removed. In particular Σ0 is
compact. Choose a complex structure on Σ which on the three cylindrical ends
pulls back to the standard structure s + it.
In order to define the appropriate trajectory space, choose three Hamiltonian
functions Hj : R × R/Z × M → R which vanish for |s| ≤ 1/2 and are independent
of s for |s| ≥ 1. Then consider the space of smooth maps
u:Σ→M
such that the restriction of u to Σ0 is a J-holomorphic curve and uj (s, t) = u◦φj (s, t)
satisfies
∂uj
∂uj
+ J(uj )
− ∇Hj (s, t, uj ) = 0
∂s
∂t
for j = 1, 2, 3. Here the gradient in the last term is to be understood with respect to
the third argument which lies in M . Any finite energy solution of these equations
satisfies the limit conditions
lim u1 (s, t) = x1 (t),
s→−∞
lim u2 (s, t) = x2 (t),
s→−∞
lim u3 (s, t) = x3 (t)
s→+∞
where xj ∈ P(Hj ). (Here we slightly abuse notation and denote by P(Hj ) the
periodic solutions of the Hamiltonian Hj (−∞, t, x) for j = 1, 2 and of Hj (+∞, t, x)
160
for j = 3.) The space of such solutions u in the correct homology class will be
denoted by M(x̃1 , x̃2 , x̃3 ). It has dimension
dim M(x̃1 , x̃2 , x̃3 ) = µ(x̃3 , H3 ) − µ(x̃1 , H1 ) − µ(x̃2 , H2 )
In the zero dimensional case we get finitely many solutions by the same argument
as above.
By the usual counting procedure, these finitely many solutions determine a chain
map
CF k (H1 ) × CF ` (H2 ) → CF k+` (H3 ) : (ξ, η) 7→ ξ ∗ η.
It follows by the usual gluing techniques in Floer homology that this map is a
cochain homomorphism and therefore induces a homomorphism of Floer cohomologies
HF k (H1 ) × HF ` (H2 ) → HF k+` (H3 ).
In view of its construction this map is called the pair-of-pants product. It is
easy to see, by the usual deformation arguments, that the pair-of-pants product is
independent of the conformal structure on the Riemann surface Σ used to define
it. It is also independent of the Hamiltonian functions Hj , as long as they are not
changed at ±∞. If they are changed at ±∞, one again uses Floer’s gluing techniques
to prove that the product is natural with respect to the isomorphisms Φβα of
Theorem 10.1.2. To prove that the product is skew-commutative with the usual
sign conventions, choose an orientation preserving diffeomorphism f : Σ → Σ which
interchanges the two cylindrical ends on the left. This diffeomorphism changes the
conformal structure of Σ but, as we have seen above, the resulting map on Floer
cohomology is independent of the choice of the conformal structure on Σ. The proof
of associativity requires the gluing of two surfaces Σ12,3 and Σ34,5 with 3 cylindrical
ends over a very long cylinder. This process cancels the two ends labelled by 3 and
results in a Riemann surface Σ124,5 with 4 cylindrical ends. If the neck is sufficiently
long then the resulting triple product corresponds to the composition (ξ ∗η)∗ζ. Now
vary the conformal structure on the surface Σ124,5 and decompose it in a different
way to obtain the identity (ξ ∗ η) ∗ ζ = ξ ∗ (η ∗ ζ). This variation of the conformal
structure corresponds to the moving of the point z in the proof of the associativity
of quantum cohomology in Section 8.2.
An alternative definition of a product structure on Floer cohomology which uses
differential forms is presented by Viterbo in [83].
10.3
A comparison theorem
According to Theorem 10.1.3 the Floer cohomology groups HF ∗ (M ) are naturally
isomorphic to the quantum cohomology groups QH ∗ (M ) and one would expect
the pair-of-pants product to correspond to the quantum deformation of the cup
product under this isomorphism. This question is already nontrivial in the case
π2 (M ) = 0 where the quantum deformation agrees with the ordinary cup product.
This was recently proved by Schwarz [77] in his thesis for the case π2 (M ) = 0,
by Piunikhin–Salamon–Schwarz [63] for the general case, and independently by
Liu [39] and Ruan–Tian [69].
10.3. A COMPARISON THEOREM
161
Theorem 10.3.1 Let Φα : HF ∗ (M, ω, H α , J α ) → QH ∗ (M ) be the isomorphisms
of Theorem 10.1.3 and similarly for Φβ and Φγ . Then
Φγ (ξ α ∗ ξ β ) = Φα (ξ α ) ∗ Φβ (ξ β )
for ξ α ∈ HF ∗ (H α ) and ξ β ∈ HF ∗ (H β ). Here the product on the left is defined
by the pair-of-pants construction while the product on the right is the quantum
deformation of the cup product.
The proof of this theorem goes along the following lines. First use the MorseWitten complex to represent the quantum deformation of the cup product in a
somewhat different way. Given three Morse functions H1 , H2 , H3 : M → R, three
critical points x1 , x2 , x3 , and three distinct points z1 , z2 , z3 ∈ CP 1 , consider the
space
MA (x1 , x2 , x3 ) = MA (x1 , x2 , x3 ; H, z)
of all J-holomorphic A-spheres u : CP 1 → M such that
u(z1 ) ∈ W u (x1 , H1 ),
u(z2 ) ∈ W u (x2 , H2 ),
u(z3 ) ∈ W s (x3 , H3 ).
Here W s (x, H) and W u (x, H) denote the stable and unstable manifolds of a critical
point x of H with respect to the (upward) gradient flow of the Morse function H.
One can think of such objects as spiked J-holomorphic spheres. Since
dim W s (x, H) = indH (x),
dim W u (x, H) = 2n − indH (x),
we have generically
dim MA (x1 , x2 , x3 ) = 2c1 (A) + indH3 (x3 ) − indH1 (x1 ) − indH2 (x2 ).
Whenever this dimension is zero, denote by nA (x1 , x2 , x3 ) the number of points in
MA (x1 , x2 , x3 ) counted with appropriate signs. This gives rise to a chain map
C ∗ (H1 ) × C ∗ (H2 ) → C ∗ (H3 )
defined by
hx1 i ∗ hx2 i =
X
nA (x1 , x2 , x3 )hx3 i,
(10.7)
A
and this induces a map on quantum cohomology. Here the boundary map is taken
to be δMW and is determined by counting the connecting orbits of the finite dimensional gradient flows. This product agrees with the deformed cup product because
the stable and unstable manifolds of critical points of Morse functions represent
cohomology classes which generate the cohomology of M .
Now it follows from the standard gluing arguments in Floer homology that
the isomorphism Φα : HF ∗ (H α ) → QH ∗ (M ) of Theorem 10.1.3, as described
in [63], intertwines the two product structures. To see this one has to glue three
J-holomorphic spiked disks to the boundary of a J-holomorphic pair-of-pants. More
precisely, the boundary components are cylindrical ends abutting on periodic solutions and one uses Floer’s gluing theorem. As a result one obtains a (perturbed)
J-holomorphic sphere with three spikes as described above. More details of this
argument are given in [63]. The full details of the proof will appear in Schwarz [78].
A physicist’s approach to this problem may be found in Sadov [72]. Note also that
Theorem 10.3.1 gives rise to an alternative proof of the associativity of quantum
cohomology.
162
10.4
Donaldson’s quantum category
Symplectomorphisms
The Floer homology approach to quantum cohomology can be extended in two
directions. Assume for simplicity that the symplectic manifold (M, ω) is compact,
simply connected, and monotone. Then there are Floer cohomology groups HF ∗ (φ)
for every symplectomorphism φ. As above they are graded modulo 2N where N is
the minimal Chern number, and the Euler characteristic of the theory
χ(HF ∗ (φ)) = L(φ)
is the Lefschetz number of φ. The critical points that generate the Floer complex are now the fixed points of φ, which we assume to be all nondegenerate, and
the connecting orbits are J-holomorphic maps u : R × [0, 1] → M which satisfy
u(s, 1) = φ(u(s, 0)). We may think of these as J-holomorphic sections of a symplectic fiber bundle P → R × S 1 with fiber M and holonomy φ around S 1 . If there
are degenerate fixed points then we must choose a Hamiltonian perturbation as in
the previous section. The resulting Floer homology groups are independent of the
Hamiltonian perturbation and the almost complex structure used to define them
and they only depend on the symplectic isotopy class of φ. (See [20], [22], and [14]
for more details.) Now for any two symplectomorphisms φ and ψ there is a natural
isomorphism
HF ∗ (φ) → HF ∗ (ψ ◦ φ ◦ ψ −1 ).
Moreover, according to Donaldson, there is an analogue of the deformed cupproduct, namely a symmetric and associative pairing
HF ∗ (φ) ⊗ HF ∗ (ψ) → HF ∗ (ψ ◦ φ).
This should be defined in terms of J-holomorphic sections of a symplectic fiber
bundle P → S with fiber M where S is a 2-sphere with three punctures and the
holonomies around these punctures are conjugate to φ, ψ, and ψ ◦ φ, respectively.
This product structure can be interpreted as a category in which the objects are
the symplectomorphisms of M and and the morphisms from φ to ψ are the elements
of the Floer cohomology group HF ∗ (ψ ◦ φ−1 ). Composition of two morphisms is
given by the quantum product.
Now in the case ψ = id it follows from the discussion of Section 10.1 that, in
the monotone case, the Floer cohomology groups are isomorphic to the ordinary
cohomology groups HF ∗ (id) = QH ∗ (M ) = H ∗ (M ), where the grading is made
periodic with period 2N . Thus the cohomology of M acts on the Floer cohomology
of φ
H ∗ (M ) ⊗ HF ∗ (φ) → HF ∗ (φ).
In this case there is yet another way to think of the cuantum product structure.
Namely, one can represent a k-dimensional cohomology class by a codimension-k
submanifold (or pseudocycle) V ⊂ M and then intersect the spaces of connecting
orbits in the construction of the Floer homology groups of φ with this submanifold.
More generally, one can intersect these connecting orbit spaces with finite dimensional submanifolds of the path space Ωφ = {γ : [0, 1] → M | γ(1) = φ(γ(0))} and
obtain an action of the (low dimensional) cohomology H ∗ (Ωφ ) on HF ∗ (φ).
10.4. DONALDSON’S QUANTUM CATEGORY
163
Specializing further to the case φ = id, we see that the above product construction agrees with the pair-of-pants product of Section 10.2, and so in this case the
Donaldson category reduces to the quantum cohomology of M .
Mapping tori
The previous structures are particularly interesting when the symplectic manifold
M is the moduli space MΣ of flat connections over a Riemann surface Σ (of large
genus) with structure group SU(2) or SO(3). In this case the mapping class group
of Σ acts on the manifold MΣ by symplectomorphisms φf : MΣ → MΣ (where f :
Σ → Σ is an orientation preserving diffeomorphism) and one can examine the Floer
cohomology groups HF ∗ (φf ) generated by the mapping class group. Results in this
direction will appear in the thesis of M. Callaghan [7]. Now there is an alternative
construction, based on Floer cohomology groups for 3-manifolds and Yang-Mills
instantons on 4-dimensional cobordisms. Every diffeomorphism f : Σ → Σ induces
a mapping torus Yf = Σ × R/ ∼ where Σ × {t} is identified with Σ × {t + 1}
via f . These 3-manifolds determine Floer homology groups HF ∗ (Yf ) constructed
from flat SO(3)-connections on Yf and anti-self-dual instantons on Yf × R. It was
conjectured by Atiyah and Floer, and proved in Dostoglou–Salamon [16], that there
is a natural isomorphism
HF ∗ (φf ) ∼
= HF ∗ (Yf ).
Now in Floer-Donaldson theory there is a pairing
HF ∗ (Yf ) ⊗ HF ∗ (Yg ) → HF ∗ (Ygf )
determined by anti-self-dual instantons over the 4-manifold X which is fibered over
the 3-punctured sphere S with fiber S and holonomy f , g, and gf . Of course, it is
natural to conjecture that the two product structures should be preserved by the
above isomorphisms and this will be proved in Salamon [74].
An interesting special case arises when g = id. In this case the symplectic Floer
cohomology of φg and hence the instanton Floer cohomology of Yg agrees with the
ordinary cohomology of the moduli space MΣ made periodic with period 4. So
H ∗ (MΣ ) acts on the Floer cohomology of Yf
H ∗ (MΣ ) ⊗ HF ∗ (Yf ) → HF ∗ (Yf ).
Now the cohomology of MΣ is well understood and is closely related to the homology
of the Riemann surface Σ itself. For example there is a universal construction of
a homomorphism µ : H1 (Σ) → H 3 (BΣ ) where BΣ = AΣ /GΣ denotes the infinite
dimensional configuration space of connections on the bundle P → Σ modulo gauge
equivalence. This is Donaldson’s µ-map. It can be roughly described as the slantproduct
1
µ(γ) = − p1 (P)/γ
4
where p1 (P) ∈ H 4 (BΣ × Σ) denotes the first Pontryagin class of the universal
SO(3)-bundle P → BΣ × Σ. In more explicit terms, the induced cohomology class
in H 3 (MΣ ) is represented by the codimension-3-submanifold Vγ ⊂ MΣ of those
164
flat connections which have trivial holonomy around γ. In summary, we have the
following diagrams
HF ∗ (Yf )
↓
µ(γ)
HF ∗ (φf )
→
→
HF ∗ (Yf )
,
↓
Vγ
HF ∗ (φf )
HF ∗ (Yf )
↓
HF ∗ (φf )
Xγ
→
→
HF ∗ (Yf )
.
↓
∗
HF (φf )
In each diagram the vertical arrows are the natural isomorphisms of [16]. In the
diagram on the left the horizontal maps are defined by cutting down the moduli
spaces of connecting orbits by intersecting them with suitable submanifolds. In
the diagram on the right the horizontal maps are given by the product structures
which are defined in terms of cobordisms. For example, the class γ ∈ H1 (Σ)
determines a natural cobordism Xγ with boundary ∂Xγ = (−Yf ) ∪ Yf . Of course,
all four definitions of the product should agree under the natural isomorphisms.
The relations between these product structures play an important role in the work
of Callaghan about symplectic isotopy problems on MΣ and a detailed discussion
will appear in his thesis [7].
These product structures also play an important role in the Floer-Fukaya construction of cohomology groups HF F ∗ (Y, γ) associated to pairs (Y, γ) where Y is
a 3-manifold and γ ∈ H1 (Y ). In another direction, the homomorphisms in Floer’s
exact sequence can be interpreted in terms of these product structures and any
analogue in symplectic Floer theory should be related to quantum cohomology.
Lagrangian intersections
There are similar structures in Floer cohomology for Lagrangian intersections.
These form in fact the original context of Donaldson’s quantum category construction. With M as above (compact, simply connected, and monotone) there are Floer
cohomology groups HF ∗ (L0 , L1 ) for every pair of Lagrangian submanifolds L0 and
L1 with H 1 (Li , R) = 0. In this case the critical points are the intersection points
L0 ∩ L1 and the connecting orbits are J-holomorphic curves u : R × [0, 1] → M with
u(s, 0) ∈ L0 and u(s, 1) ∈ L1 . This is the context of Floer’s original work in [18].
The Euler charcteristic of Floer cohomology is now the intersection number
χ(HF ∗ (L0 , L1 )) = L0 · L1 .
Again there is a pairing
HF ∗ (L0 , L1 ) ⊗ HF ∗ (L1 , L2 ) → HF ∗ (L0 , L2 )
defined by holomorphic triangles. In [18] Floer proved that HF ∗ (L0 , L0 ) is isomorphic to the ordinary cohomology of L0 provided that π2 (M ) = 0. Under our
assumptions, where M is simply connnected, this condition is never satisfied and
the corresponding assertion is an open question.
This multiplicative structure can be interpreted as a quantum category where
the objects are the Lagrangian submanifolds L ⊂ M (with H 1 (L, R) = 0) and
the morphisms from L0 to L1 are the elements of the Floer cohomology group
HF ∗ (L0 , L1 ). The above structur with symplectomorphisms is a special case of
this with M = N × N , L0 = ∆, L1 = graph (φ), and L2 = graph (ψ).
10.5. CLOSING REMARK
165
Heegard splittings
The Floer theory for Lagrangian intersections is related to 3-manifolds as follows.
If Y is a homology-3-sphere choose a Heegard splitting Y = Y0 ∪ Y1 over a Riemann
surface Σ. Then each handlebody Yj determines a Lagrangian submanifold Lj =
LYj ⊂ MΣ and, according to Atiyah [4] and Floer, there should be a natural
isomorphism
HF ∗ (Y0 ∪ Y1 ) ∼
= HF ∗ (L0 , L1 ).
As before, there is a product structure
HF ∗ (Y0 ∪ Y1 ) ⊗ HF ∗ (Y1 , Y2 ) → HF ∗ (Y0 , Y2 )
defined directly in terms of Yang-Mills instantons on a suitable 4-dimensional cobordism and the two product structures should be related by the above (conjectural)
isomorphisms.
10.5
Closing remark
We close this book with the observation that both in the definition of Floer cohomology and in the definition of quantum cohomology the same difficulties arise
from the presence of J-holomorphic spheres with negative Chern number. In order
to extend either theory to general compact symplectic manifolds one has to find
techniques of dealing with J-holomorphic curves of negative Chern number and
their multiple covers. For example an important putative theorem would be that
multiply covered J-holomorphic curves of negative Chern number cannot be approximated by simple curves (in the same homology class). In the almost complex
case no one has so far developed such techniques. One would expect that if such
methods can be found then they should give rise to both an extension of quantum
cohomology to general compact symplectic manifolds and a proof of the Arnold
conjecture in the general case.
166
Appendix A
Gluing
In this section we give a proof of the decomposition formula in Lemma 8.2.5. This
proof is based on a gluing theorem for J-holomorphic curves which is the converse of
Gromov’s compactness theorem. It asserts, roughly speaking, that if two (or more)
J-holomorphic curves intersect and satisfy a suitable transversality condition then
they can be approximated, in the weak sense of Section 4.4, by a sequence of Jholomorphic curves representing the sum of their homology classes. The proof is
along similar lines as the Taubes gluing theorem for anti-self-dual instantons on 4manifolds. Our proof is an adaption of the argument in Donaldson–Kronheimer [13],
page 287–295, to the two dimensional case of J-holomorphic curves.
We shall consider two J-holomorphic curves u ∈ M(A, J) and v ∈ M(A, J)
such that u(0) = v(∞) and the corresponding Fredholm operators Du and Dv are
surjective. This means that the moduli spaces M(A, J) and M(B, J) are smooth
manifolds near u and v, respectively. However, as pointed out to us by Gang Liu,
this condition alone cannot be enough to obtain a nearby J-holomorphic curve
w with surjective Fredholm operator Dw . Consider for example the case where
c1 (A) < 0 and c1 (B) < 0 with index Du = 2n + 2c1 (A) > 0 and index Dv =
2n + 2c1 (B) > 0 but index Dw = 2n + 2c1 (A + B) < 0. Then there cannot be
any J-holomorphic curve in the class A + B with surjective Fredholm operator. To
obtain such a curve we must assume that
2n + 2c1 (A) + 2c1 (B) > 0.
In fact we shall assume that the evaluation map
M(A, J) × M(B, J) → M × M : (u, v) 7→ (u(0), v(∞))
is transverse to the diagonal in M ×M . This implies the above inequality. Moreover,
by Theorem 6.3.2, this is satisfied for generic J.
Given two curves u ∈ M(A, J) and v ∈ M(B, J) with u(0) = v(∞) we shall
then consider an approximate J-holomorphic curve u#v and prove that nearby
there is an actual J-holomorphic curve w. In particular, this involves a proof that
the Fredholm operator Du#v is onto and an estimate for the right inverse. For
this we shall need the following elementary, but subtle, observations about cutoff
functions. Throughout the appendix all integrals are to be understood with respect
to the Lebesgue measure unless otherwise mentioned.
167
168
APPENDIX A. GLUING
A.1
Cutoff functions
Lemma A.1.1 For every constant ε > 0 there exists a δ > 0 and a smooth cutoff
function β : R2 → [0, 1] such that
1, if |z| ≤ δ,
β(z) =
0, if |z| ≥ 1,
and
Z
|∇β(z)|2 ≤ ε.
|z|≤1
In fact, such a function exists with δ = e−2π/ε .
Proof: We first define a cutoff function of class W 1,2 by β(z) = 1 for |z| ≤ δ,
β(z) = 0 for |z| ≥ 1, and
β(z) =
log |z|
,
log δ
δ ≤ |z| ≤ 1.
Then the gradient ∇β satisfies
|∇β(z)| =
1
.
|z| · | log δ|
Hence
Z
|∇β(z)|2 =
δ≤|z|≤1
Z
δ≤|z|≤1
1
=
2
|z| | log δ|2
Z
δ
1
2π
2π
ds =
.
2
s| log δ|
| log δ|
This function β is only of class W 1,2 but not smooth. To obtain a smooth function
take the convolution with φN (z) = N 2 φ(N z) where N is large and φ : R2 → R is
any smooth function with support in the unit ball and mean value 1.
2
The previous lemma says that there is a sequence of compactly supported functions on R2 which converge to zero in the W 1,2 norm but not in the L∞ -norm. This
is a borderline case for the Sobolev estimates. For p > 2 there is an embedding
W 1,p (R2 ) ,→ C(R2 ) and hence the assertion of the previous lemma does not hold if
we replace the L2 -norm of ∇β by the Lp -norm for p > 2. In the following we shall
see that the L2 -norm is precisely what is needed for the proof of surjectivity of a
first order operator in 2 dimensions. A similar argument in n dimensions requires
the Ln -norm of ∇β to be small and this is again a Sobolev borderline case. The
case n = 4 is relevant for the gluing of anti-self-dual instantons and this is explained
in [13], page 287–295.
Now consider the functions
βλ (z) = β(λz)
where β is as in Lemma A.1.1 and λ ≥ 1. In view of the conformal invariance of
the L2 -norm of ∇β we have the following estimate for q < 2 and ξ ∈ W 1,2 (R2 )
k∇βλ · ξkLq ≤ k∇βλ kL2 kξkLr ≤ c k∇βλ kL2 kξkW 1,q .
A.1. CUTOFF FUNCTIONS
169
The first inequality above uses the Hölder inequality
Z
q
|uv|
1/q
Z
≤
|u|
s
1/s Z
|v|
r
1/r
1 1
1
+ = ,
s r
q
,
with s = 2. Hence we must take r = 2q/(2 − q) and so, by Theorem B.1.5, there is
a Sobolev embedding W 1,q ,→ Lr . This implies the second inequality above. If the
L2 -norm of ∇β is sufficiently small then we obtain
k∇βλ · ξkLq ≤ ε kξkW 1,q
for every ξ ∈ W 1,q (R2 ). In the above proof of this inequality it is essential to
assume that q < 2. However, the following lemma shows that the last inequality
remains valid for q > 2 provided that ξ(0) = 0.
Lemma A.1.2 For every p > 2 and every ε > 0 there exists a δ > 0 and a smooth
cutoff function β : R2 → [0, 1] as in Lemma A.1.1 such that
k∇βλ · ξkLp ≤ ε kξkW 1,p
for every ξ ∈ W 1,p (R2 ) with ξ(0) = 0 and every λ ≥ 1. Here βλ : R2 → [0, 1] is
defined by βλ (z) = β(λz).
Proof: With β as in the proof of Lemma A.1.1
|∇βλ (z)| = λ|∇β(λz)| =
1
.
|z| · | log δ|
Moreover, by Theorem B.1.4, ξ is Hölder continuous with exponent µ = 1 − 2/p
and hence there exists a constant c > 0 (depending only on p) such that
|ξ(z)| ≤ c|z|1−2/p kξkW 1,p
for every ξ ∈ W 1,p (R2 ) with ξ(0) = 0. Hence
Z
|∇βλ (z)|p |ξ(z)|p
Z
1
≤
δ/λ≤|z|≤1/λ
δ/λ≤|z|≤1/λ
≤
cp
|z|p | log δ|p
Z
δ/λ≤|z|≤1/λ
≤
cp
Z
1/λ
δ/λ
=
cp
p
cp |z|p−2 kξkW 1,p
1
p
kξkW 1,p
|z|2 | log δ|p
2π
p
ds kξkW 1,p
s| log δ|p
2π
p
kξkW 1,p .
| log δ|p−1
This proves the lemma with β ∈ W 1,∞ . To obtain a smooth cutoff function one
uses the usual convolution argument.
2
170
A.2
APPENDIX A. GLUING
Connected sums of J-holomorphic curves
Fix a regular almost complex structure J such that the operator Du is onto for
every J-holomorphic curve u : CP 1 → M . Consider the space
M(A, B, J) ⊂ M(A, J) × M(B, J)
of intersecting pairs (u, v) of J-holomorphic curves which represent the classes
A, B ∈ H2 (M ) and satisfy
u(0) = v(∞).
Denote by MK (A, B, J) ⊂ M(A, B, J) the subset of those pairs (u, v) which satisfy
kdukL∞ ≤ K and kdvkL∞ ≤ K, where the norm is taken with respect to the FubiniStudy metric on CP 1 and the J-induced metric (3.1) on M .
Given a pair (u, v) ∈ MK (A, B, J) and a large number R > 0 we shall construct
an approximate J-holomorphic curve wR : CP 1 → M which (approximately) agrees
with u on the complement of a disc B1/R of radius 1/R and with the rescaled curve
v(R2 z) on B1/R . The condition u(0) = v(∞) guarantees that these maps are
approximately equal on the circle |z| = 1/R. We shall thicken this circle on each
side to annuli of the form A(δr, r) for a sufficiently small number δ > 0. We shall
use cutoff functions as constructed in Lemma A.1.1 which are supported in these
annuli and this determines the smallness of δ. We shall fix a suitable constant δ
and consider sufficiently large numbers R > Rδ so that the product δR is large. In
the limit R → ∞ we obtain the converse of Gromov’s compactness. In other words
the curves wR will converge in the weak sense of Section 4.4 to the pair (u, v).
More precisely, for any sufficiently small number δ > 0, any sufficiently large
number R > 0, and any pair (u, v) ∈ MK (A, B, J) we shall construct an approximate J-holomorphic curve wR = u#R v : CP 1 → M which satisfies

δ
,
v(R2 z), if |z| ≤ 2R





1
wR (z) =
u(0) = v(∞), if Rδ ≤ |z| ≤ δR
,





2
u(z), if |z| ≥ δR
.
To define the function wR in the rest of the annulus δ/2R ≤ |z| ≤ 2/δR we fix a
cutoff function ρ : C → [0, 1] such that
1, if |z| ≥ 2,
ρ(z) =
0, if |z| ≤ 1.
Now denote the point of intersection of the curves u and v by x = u(0) = v(∞)
and use the exponential map in a neighbourhood of this point. Let ξu (z) ∈ Tx M
for |z| < ε and ξv (z) ∈ Tx M for |z| > 1/ε be the vector fields such that u(z) =
expx (ξu (z)) and v(z) = expx (ξv (z)). Define
wR (z) = expx ρ(δRz)ξu (z) + ρ(δR/z)ξv (R2 z)
for δ/2R ≤ |z| ≤ 2/δR. This is well defined whenever R > 2/δε. Moreover, the
number ε > 0 depends only on the uniform bound K for du and dv. The map wR
is not a J-holomorphic curve, however we shall see that the error converges to zero
as R → ∞. Our goal is construct a true J-holomorphic curve w̃R near wR .
A.3. WEIGHTED NORMS
171
Remark A.2.1 The curves wR converge to the pair (u, v) in the weak sense of
Section 4.4. More precisely, wR (z) converges to u(z) uniformly with all derivatives
on compact subsets of CP 1 − {0} and wR (z/R2 ) converges to v(z) uniformly with
all derivatives on compact subsets of C = CP 1 − {∞}. The J-holomorphic curves
w̃R which we construct below will converge to the pair (u, v) in the same way. 2
Remark A.2.2 Consider the curves uR , vR : CP 1 → M defined by
wR (z), if |z| ≥ 1/R,
uR (z) =
u(0), if |z| ≤ 1/R,
and
vR (z) =
wR (z/R2 ),
v(∞),
if |z| ≤ R,
if |z| ≥ R.
Then uR converges to u in the W 1,p -norm and vR converges to v in the W 1,p -norm.
Moreover, the convergence is uniform over all pairs (u, v) ∈ MK (A, B, J). Note
that uniformity requires the uniform estimate on the derivatives of du and dv (by
the constant K). Note also that uR and vR do not converge in the C 1 -norm.
2
Example A.2.3 Consider the case M = CP 1 with the standard complex structure. Then the holomorphic curves
u(z) = 1 + z,
satisfy u(0) = v(∞) = 1 as required. The

 1 + 1/R2 z,
1,
wR (z) =

1 + z,
v(z) = 1 + 1/z
above maps wR : CP 1 → CP 1 satisfy
if |z| ≤ δ/2R,
if δ/R ≤ |z| ≤ 1/δR,
if |z| ≥ 2/δR.
Nearby J-holomorphic curves are given by
w̃R (z) = v(R2 z) + u(z) − 1 =
z 2 + z + 1/R2
z
and these converge to the pair (u, v) as in Remark A.2.1.
A.3
2
Weighted norms
In order to apply the implicit function theorem we must specify the norm in which
∂¯J (wR ) is small. The guiding principle for our choice of norm is the observation
that the curves u and v play equal roles in this gluing argument. However, the map
v appears in rescaled form and is concentrated in a small ball of radius 1/R. So in
order to give u and v equal weight we shall consider a family of R-dependent metrics
on the 2-sphere such that the volume of the ball of radius 1/R is approximately
equal to the volume of S 2 with respect to the standard (Fubini-Study) metric. The
−2
rescaled metric is of the form θR
(ds2 + dt2 ) where
−2
R + R2 |z|2 , if |z| ≤ 1/R,
θR (z) =
1 + |z|2 , if |z| ≥ 1/R.
172
APPENDIX A. GLUING
The area of S 2 with respect to this metric is given by
Z
−2
2
VolR (S ) =
θR
≤ 2π.
(A.1)
C
where, as always, we integrate with respect to Lebesgue measure. The effect of
this metric on the Lp and W 1,p norms of vector fields along wR is as follows. If we
rescale the vector field ξ(z) ∈ TwR (z) M with support in B1/2R to obtain a vector
field ξ(z/R2 ) ∈ Tv(z) M along v then the standard norms of this rescaled vector
field agree with the weighted norms of the original vector field ξ. More explicitly,
define the weighted norms
Z
kξk0,p,R =
θR (z)−2 |ξ(z)|p
1/p
,
C
Z
kξk1,p,R =
θR (z)
−2
p
|ξ(z)| + θR (z)
p−2
|∇ξ(z)|
p
1/p
.
C
Here ∇ denotes the Levi-Civita connection of the metric induced by J and, with
z = s + it, we denote |∇ξ(z)|2 = |∇s ξ(z)|2 + |∇t ξ(s)|2 . Similarly, for 1-forms
∗
η = η1 ds + η2 dt ∈ Ω1 (wR
T M ), consider the norms
Z
kηk0,p,R =
θR (z)p−2 |η(z)|p
1/p
,
C
kηk0,∞,R = sup θR (z)|η(z)|,
z∈C
Z
kηk1,p,R =
θR (z)p−2 |η(z)|p + θR (z)2p−2 |∇η(z)|p
1/p
.
C
where |η|2 = |η1 |2 + |η2 |2 and |∇η|2 = |∇s η1 |2 + |∇t η1 |2 + |∇s η2 |2 + |∇t η2 |2 .
In the case R = 1 these are the usual Lp and W 1,p norms with respect to the
Fubini-Study metric on C (as a coordinate patch of CP 1 ). For general R these
norms should be considered in two parts. In the domain |z| ≥ 1/R they are still
the usual norms and in the domain |z| ≤ 1/R they agree with the usual norms of
the rescaled vector field
˜ = ξ(R−2 z)
ξ(z)
or 1-form
η̃ = R−2 (η1 (R−2 z)ds + η2 (z/R2 )dt)
along v(z) in the ball of radius R, again with respect to the Fubini-Study metric on
C. Hence we obtain the usual Sobolev estimates (see Theorems B.1.4 and B.1.5)
with constants which are independent of R.
Lemma A.3.1 (i) For p > 2 and K > 0 there exist constants R0 > 0 and c > 0
such that
kξkL∞ ≤ c kξk1,p,R ,
kηk0,∞,R ≤ c kηk1,p,R
∗
∗
for all R ≥ R0 , ξ ∈ C ∞ (wR
T M ), η ∈ Ω1 (wR
T M ), (u, v) ∈ MK (A, B, J).
A.4. AN ESTIMATE FOR THE INVERSE
173
(ii) For p < 2, 1 ≤ q ≤ 2p/(2 − p), and K > 0 there exist constants R0 > 0 and
c > 0 such that
kξk0,q,R ≤ c kξk1,p,R ,
kηk0,q,R ≤ c kηk1,p,R
∗
∗
for all R ≥ R0 , ξ ∈ C ∞ (wR
T M ), η ∈ Ω1 (wR
T M ), (u, v) ∈ MK (A, B, J).
The following lemma can be proved by a straightforward argument involving
change of variables. It plays a key role in our application of the implicit function
theorem. Note in fact that in Theorem 3.3.4 we do not specify the metric on the
Riemann surface Σ = CP 1 with respect to which the norms are defined. We do
however assume that the volume and the Lp -norm of du are uniformly bounded
with respect to this metric.
Lemma A.3.2 For p > 2 and K > 0 there exist constants R0 > 0 and c > 0 such
that
kdwR k0,p,R ≤ c
for R ≥ R0 and (u, v) ∈ MK (A, B, J).
A.4
An estimate for the inverse
We must now prove that the Fredholm operator DwR is surjective with a uniformly
bounded right inverse provided that R > 0 is sufficiently large. This argument will
involve the cutoff function of Lemma A.1.1. We use the notation of Section 3.3.
We begin by rephrasing our transversality condition. Abbreviate
1,p
Wu,v
= (ξu , ξv ) ∈ W 1,p (u∗ T M ) × W 1,p (v ∗ T M ) | ξu (0) = ξv (∞)
and
Lpu = Lp (Λ0,1 T ∗ CP 1 ⊗J u∗ T M ).
1,p
Note that this definition of the space Wu,v
only makes sense for p > 2 since it is only
in this case that W 1,p -sections are continuous and can be evaluated at a point. The
1,p
∗
space Wu,v
should be interpreted as a limit as R → ∞ of the spaces W 1,p (wR
TM)
of vector fields along the glued curves. Our transversality assumption means that
the operator
1,p
Du,v : Wu,v
→ Lpu × Lpv ,
Du,v (ξu , ξv ) 7→ (Du ξu , Dv ξv )
1,p
is onto. Hence Du,v has a right inverse Qu,v : Lpu × Lpv → Wu,v
such that
Du,v ◦ Qu,v = 1l,
kQu,v k ≤ c0 .
Here the constant c0 can be chosen independent of the pair (u, v) ∈ MK (A, B, J)
but will in general depend on A, B, K, and J. Because the space MK (A, B, J)
is compact it suffices to prove this locally and this can be done by reducing the
dimension of the kernel of Du,v to zero through additional (say pointwise) conditions
on the pair (ξu , ξv ).
174
APPENDIX A. GLUING
Remark A.4.1 Here are a few more comments on the existence of the uniformly
bounded inverse Qu,v of the operator Du,v .
1. Assume first that Du,v has index zero. Then it is bijective and the open
−1
mapping theorem shows that Qu,v = Du,v
is bounded. Moreover, the operator
Du,v depends continuously on (u, v) in the norm topology. This implies that also
the operator Qu,v depends continuously on (u, v) in the norm topology and hence
the required estimate is uniform in the case of index zero. (Warning: Note that the
domains of the operators and their range actually depend on the pair (u, v). So one
1,p
has to choose some kind of identification of say, Wu,v
and Wu1,p
0 ,v 0 for nearby pairs
0 0
(u, v) and (u , v ) to make this continuous dependence precise. This can be done
by parallel transport along short geodesics as in the proof of Theorem 3.3.4.)
2. If the index is bigger than zero, for example 2n, one can fix some point
z0 ∈ S 2 and impose the condition ξu (z0 ) = 0. For a generic point z0 this cuts
down the dimension of the kernel by 2n and results in an operator of index zero,
reducing to the previous case. More generally cut down by imposing the condition
ξu (z0 ) ∈ Eu(z0 ) for some subbundle E ⊂ T M of the right dimension. In any case,
reduce to the case of index zero to deal with the general case.
2
Our strategy is now to use the operator Qu,v to construct, for R sufficiently
∗
T M ) of the operator
large, an approximate right inverse QR : LpwR → W 1,p (wR
DwR such that
kQR k ≤ c1 ,
kDwR QR − 1l k < 1/2.
(A.2)
Under these conditions the operator DwR QR : LpwR → LpwR is invertible and a right
inverse of DwR is given by
QwR = QR (DwR QR )−1 .
Roughly speaking, we will construct the operator QR by means of the commutative
diagram
1,p
Wu,v
↓
W
1,p
∗
(wR
TM)
Qu,v
←−
Lpu × Lpv
↑
QR
LpwR .
←−
(A.3)
Here the vertical maps are given in terms of cutoff functions. In fact it is convenient
to modify the diagram slightly and replace u and v by the curves uR and vR defined
in Remark A.2.2. Then uR converges to u in the W 1,p -norm and similarly for vR .
Hence the operator DuR ,vR still has a uniformly bounded right inverse QuR ,vR . The
following diagram should serve as a guide to the definition of QR .
(ξu , ξv )
↓
ξ
QuR ,vR
←−
QR
←−
(ηu , ηv )
↑
η.
Given η ∈ LpwR we define the pair (ηu , ηv ) ∈ Lpu × Lpv ) simply by cutting off η along
the circle |z| = 1/R:
−2
ηR (z), if |z| ≥ 1/R,
R ηR (R−2 z), if |z| ≤ R,
ηu (z) =
ηv (z) =
0, if |z| ≤ 1/R,
0, if |z| ≥ R.
A.4. AN ESTIMATE FOR THE INVERSE
175
The discontinuities in ηu and ηv do not cause problems because only their Lp -norms
enter the estimates. Now the pair (ξu , ξv ) is defined in terms of the right inverse
QuR ,vR by
(ξu , ξv ) = QuR ,vR (ηu , ηv ).
In particular
ξu (0) = ξv (∞) = ξ0 ∈ Tu(0) M.
Finally, we define ξ = QR η by

ξu (z),








 ξu (z) + (1 − β(1/Rz))(ξv (R2 z) − ξ0 ),
ξ(z) =


ξv (R2 z) + (1 − β(Rz))(ξu (z) − ξ0 ),







ξv (R2 z),
if |z| ≥
1
δR ,
if
1
R
≤ |z| ≤
1
δR ,
if
δ
R
≤ |z| ≤
1
R,
if |z| ≤
δ
R.
The easiest way to understand this definition is by considering the case ξ0 = 0. Then
in the annulus δ/R ≤ |z| ≤ 1/δR the maps uR , vR and wR all take the constant
value x, and the function ξ(z) ∈ Tx M is simply the superposition of the functions
(1 − β(Rz))ξu (z) and (1 − β(1/Rz))ξv (R2 z). The important fact is that, in the
first term, the cutoff function β(Rz) only takes effect in the region |z| ≤ 1/R where
DuR ξu = 0. Similarly for the second term because the construction is completely
symmetric in u and v. In the case ξ0 6= 0 the formula can be interpreted in the
same way, but relative to the “origin” ξ0 .
Exercise A.4.2 Check the symmetry in the formula for ξ by considering the functions ũ(z) = v(1/z), ṽ(z) = u(1/z), w̃R (z) = wR (1/R2 z) and the vector fields
˜ = ξ(1/R2 z).
ξ˜u (z) = ξv (1/z), ξ˜v (z) = ξu (1/z), ξ(z)
Lemma A.4.3 The operator QR satisfies the estimate (A.2).
Proof: We must prove that
kDwR ξ − ηk0,p,R ≤
1
kηk0,p,R .
2
(A.4)
Since DuR ξu = ηu and DvR ξv = ηv the term on the left hand side vanishes for
|z| ≥ 1/δR and for |z| ≤ δ/R. In view of the symmetry of our formulae it suffices
to estimate the left hand side in the annulus
δ
1
≤ |z| ≤ .
R
R
In this region, uR = vR = wR is the constant map. Therefore over this annulus the
corresponding operators DuR , DvR , and DwR are all equal, and on functions are sim¯
ply the usual ∂-operator.
Further, the definition of ξv implies that DwR ξv (R2 ·) = η.
Hence, with the notation βR (z) = β(Rz),
DwR ξ − η
=
DuR ((1 − βR )(ξu − ξ0 ))
¯ R ⊗ (ξu − ξ0 )
(1 − βR )DuR (ξu − ξ0 ) + ∂β
¯
= (βR − 1)DuR ξ0 + ∂βR ⊗ (ξu − ξ0 ).
=
176
APPENDIX A. GLUING
Here we have used the crucial fact that DuR ξu = ηu = 0 in the region |z| ≤ 1/R.
Now we must estimate the Lp -norm of this 1-form with respect to the Rdependent metric. The next crucial point to observe is that the weighting function
for 1-forms is θR (z)p−2 . Since p > 2 and θR (z) ≤ θ1 (z) ≤ 2 in the region |z| ≤ 1/R,
it follows that the (0, p, R)-norm of our 1-form is smaller that the ordinary Lp -norm
(up to a universal factor less than 2). Hence we obtain the inequality
kDwR ξ − ηk0,p,R;B1/R
≤
2 kDwR ξ − ηkLp (B1/R )
≤
¯ R ⊗ (ξu − ξ0 ) p
2 kDuR ξ0 kLp (B1/R ) + 2 ∂β
L (B
1/R )
−2/p
≤
c2 R
|ξ0 | + ε kξu − ξ0 kW 1,p
≤
c3 (ε + R−2/p ) (kηu kLp + kηv kLp )
=
c3 (ε + R−2/p ) kηk0,p,R
The third inequality follows from the fact that the term DuR ξ0 can be pointwise
estimated by |ξ0 | and the Lp -norm is taken over an area at most π/R2 . Moreover, we
have used Lemma A.1.2. The fourth inequality follows from the uniform estimate
for the right inverse QuR ,vR of DuR ,vR . The last equality follows from the definition
of ηu and ηv .
In view of the symmetry, spelled out in Exercise A.4.2, we have a similar estimate
in the domain |z| ≥ 1/R and this proves that the operator QR satisfies the second
inequality in (A.2) provided that R ≥ Rδ is sufficiently large. The first inequality
is an easy exercise which can be safely left to the reader.
2
Lemma A.4.4 For every constant 1 ≤ p < ∞ there exist constants R0 > 0 and
c > 0 such that
∂¯J (wR )
≤ c(δR)−2/p
0,p,R
for R ≥ Rδ . Moreover, given a number K > 0 we may choose the same constants
Rδ and c for all pairs (u, v) ∈ MK (A, B, J).
Proof: Use the fact that wR (z) = uR (z) for |z| ≥ 1/R and that, in view of
Remark A.2.2, uR converges to the J-holomorphic curve u in the W 1,p -norm. For
|z| ≤ 1/R one can again use the symmetry of Exercise A.4.2. To obtain the more
precise estimate of the convergence rate recall that uR (z) = expx (ρ(δRz)ξu (z)).
Hence the term ∂¯J (wR ) is supported in the annulus B2/δR − B1/δR and in this
region can be expressed as a sum of two terms. One involves the first derivatives
of the cutoff function ρ(δRz), which grow like δR, multiplied by the function ξu (z)
which is bounded by a constant times 1/δR provided that |z| ≤ 2/δR. The second
term involves the first derivatives of ξu (z) and these are again uniformly bounded.
Since the integral is over a region of area at most 4π(δR)−2 this proves the lemma.
2
A.5
Gluing
The Lemmata A.3.2, A.4.4 and A.4.3 imply that for R sufficiently large the smooth
map wR : CP 1 → M satisfies the requirements of the implicit function theorem 3.3.4. Hence it follows from Theorem 3.3.4 that, for R > 0 sufficiently large,
A.5. GLUING
177
∗
there exists a unique smooth section ξR = QwR ηR ∈ C ∞ (wR
T M ) such that the
map
w̃R (z) = expwR (z) (ξR (z))
is a J-holomorphic curve and
kξR kW 1,p ≤ cR−2/p .
Thus we have constructed, for every sufficiently large real number R, a smooth map
fR : MK (A, B, J) → M(A + B, J)
given by
fR (u, v) = w̃R .
This construction works for R > R0 where R0 depends on A, B, and K. The
following lemma makes precise the sense in which the above gluing construction is
the converse of Gromov compactness. The proof is obvious, except perhaps for the
last statement which can be verified with the methods in the proof of Theorem 4.4.3.
Lemma A.5.1 As R → ∞ the J-holomorphic curves w̃R = fR (u, v) converge
weakly to the pair (u, v) in the sense of Section 4.4. More precisely, w̃R converges
on compact subsets of CP 1 − {0} to u, w̃R (R2 z) converges on compact subsets of
C = CP 1 − {∞} to v, and
|zR | → 0, |zR |R2 → ∞
=⇒
w̃R (zR ) → u(0) = v(∞).
We shall now discuss the properties of the gluing map fR for a fixed value of
R. Recall from the discussion after Theorem 5.2.1 that the space M(A, B, J) is a
smooth manifold whenever the evaluation map
M(A, J) × M(B, J) → M 2 : (u, v) 7→ (u(0), v(∞))
is transverse to the diagonal. Moreover its dimension is dim M(A, B, J) = 2n +
2c1 (A + B) and agrees with that of M(A + B, J). It follows from the uniqueness result in Proposition 3.3.5 that fR is a diffeomorphism between open sets in
M(A, B, J) and M(A + B, J). Moreover, the map fR is orientation preserving. To
see this examine the proof of Lemma A.4.3 and use a similar cutoff argument to
find an isomorphism
FR (u, v) : {(ξu , ξv ) ∈ ker Du ⊕ ker Dv | ξu (0) = ξv (∞)} → ker DwR .
This linear isomorphism mimics the nonlinear gluing construction. One first uses
∗
a cutoff function to obtain a section of the bundle wR
T M which is approximately
in the kernel of DwR and then projects orthogonally onto the kernel of DwR . The
resulting isomorphism represents (approximately) the differential of the map fR at
the point (u, v) and it can easily be seen to be orientation preserving. We summarize
our findings in the following
Theorem A.5.2 The above map fR : MK (A, B, J) → M(A + B, J) is an orientation preserving diffeomorphism onto an open set in M(A + B, J).
178
APPENDIX A. GLUING
Remark A.5.3 Note that the image of fR has compact closure in M(A + B, J)
and so is not a full neighbourhood of the boundary cusp-curves. The situation may
be explained as follows. There is an 8-dimensional reparametrization group acting
on M(A, B, J) and a 6-dimensional group acting on M(A+B, J). In MK (A, B, J)
we can think of the variation of the points 1 and ∞ in the domain of u and the
variation of 0 in the domain of v as corresponding to this 6-dimensional group
acting on M(A + B, J). Theorem A.5.2 shows that the “extra” variation of 1 in
the domain of v is translated by fR into a 2-dimensional family of geometrically
distinct (A + B)-curves. Thus for each pair (u, v) ∈ MK (A, B, J) the map fR takes
the annulus
{(u, v ◦ φ) ∈ MK (A, B, J) | φ(0) = 0, φ(∞) = ∞}
diffeomorphically onto an annulus in the space of unparametrized (A + B)-curves,
which surrounds the cusp-curve (u, v) in the same way that a large annulus {r1 <
|z| < r2 } surrounds ∞ in C. Here φ plays the role of Floer’s gluing parameter.
Proof of Lemma 8.2.5: We must prove that for R sufficiently large there is
a one-to-one correspondence (preserving intersection numbers) of pairs (u, v) ∈
M(A − B, B, J) such that
u(∞) ∈ α1 ,
u(1) ∈ α2 ,
v(0) ∈ α3 ,
v(1) ∈ α4 ,
(A.5)
w(1/R2 ) ∈ α4 .
(A.6)
with curves w ∈ M(A, J) such that
w(∞) ∈ α1 ,
w(1) ∈ α2 ,
w(0) ∈ α3 ,
The strategy is to prove first that every pair (u, v) gives rise to a suitable w = wR
for R sufficiently large and then to show, by a compactness argument, that every
w is of this form.
P
Under our dimension assumption j deg(αj ) = 6n − 2c1 (A) the codimension of
the pseudo-cycle
α = α1 × α2 × α3 × α4
is 2n + 2c1 (A) and hence agrees with that of the moduli space M(A − B, B, J).
As we have seen above, there are only finitely many pairs (u, v) ∈ M(A − B, B, J)
which satisfy (A.5), and so we may choose K so large that all these pairs are
contained in MK (A − B, B, J). Now consider the 4-fold evaluation map
eR : M(A, J) → M 4 ,
eR (w) = (w(∞), w(1), w(0), w(1/R2 ))
and note that
eR (u#R v) = (u(∞), u(1), v(0), v(1)).
By Lemma A.5.1, fR (u, v) is so close to the curve u#R v that the composition
eR ◦ fR : MK (A − B, B, J) → M 4
converges in the C ∞ -topology to the evaluation map
e : MK (A − B, B, J) → M 4 ,
e(u, v) = (u(∞), u(1), v(0), v(1)).
Since e is transverse to α so is eR ◦ fR for R sufficiently large, and so eR ◦ fR must
have precisely as many intersection points with α as the limit map e. Thus we have
A.5. GLUING
179
proved that near every solution (u, v) ∈ M(A − B, B, J) of (A.5) there is a solution
w ∈ M(A, J) of (A.6), provided that R > 0 is sufficiently large. Since the local
diffeomorphism fR is orientation preserving, the curve w contributes with the same
intersection number as (u, v).
Here we have assumed that 0 6= B 6= A. If B = 0 then v is constant and u
represents the class A with
u(∞) ∈ α1 ,
u(1) ∈ α2 ,
u(0) ∈ α3 ∩ α4 .
Moreover, the evaluation map e∞ : M(A, J) → M 4 defined by
e∞ (u) = (u(∞), u(1), u(0), u(0))
is tranverse to α = α1 × α2 × α3 × α4 . Since eR converges to e∞ in the C ∞ -topology
we have a nearby curve w ∈ M(A, J) with eR (w) ∈ α. A similar argument applies
to the case B = A.
Now let (uj , vj ) for j = 1, . . . , N denote the finite set of all pairs (u, v) ∈
M(A − B, B, J) which satisfy (A.5) for all choices of B, including B = 0 and
B = A. Denote by wj,R ∈ M(A, J) the solution of (A.6) which corresponds to the
pair (uj , vj ). We shall use Gromov compactness to show that, for R sufficiently
large, every curve w ∈ M(A, J) with eR (w) ∈ α must be one of the curves wj,R .
Suppose, by contradiction, that there exists a sequence Rν → ∞ and a sequence
wν ∈ M(A, J) which satisfies (A.6) with R = Rν but does not agree with any of the
solutions wj,Rν . Then it follows from Proposition 3.3.5 that there exists a constant
ε > 0 such that
sup dist(wν (z), wj,Rν (z)) ≥ ε
(A.7)
z∈CP 1
for all j and ν. On the other hand, by Gromov’s compactness we may assume,
passing to a subsequence if necessary, that wν converges in the weak sense of Section 4.4 to a cusp curve. Following the reasoning of Section 8.2 we see that the limit
curve has only two components and must be one of the pairs (uj , vj ). Assume first
that both components are nonconstant. Then the bubbling must occur at z = 0.
Hence wν converges on compact subsets of CP 1 − {0} to uj , and converges, after
rescaling, on compact subsets of C = CP 1 − {∞} to vj . Examining the values of
wν (0) and wν (1/Rν2 ) we see that the rescaling factor must be Rν2 . Moreover, the
argument in the proof of Theorem 4.4.3 (which shows that u(0) = v(∞)) can be
used to prove that
|zν | → 0, |zν |Rν2 → ∞
=⇒
wν (zν ) → uj (0) = vj (∞).
By construction, the sequence wj,Rν has exactly the same properties and hence wν
and wj,Rν must be arbitrarily close in the C 0 -topology provided that ν is sufficiently
large. This contradicts (A.7). The extreme cases where either uj or vj are constant
can be safely left to the reader. This proves the lemma and hence Theorem 8.2.1.
2
Remark A.5.4 (i) Lemma 8.2.5 can be proved along a slightly different route,
namely by first taking the approximate J-holomorphic curve wj = uj #R vj :
CP 1 → M obtained from the gluing construction. This map satisfies eR (wj ) ∈
180
APPENDIX A. GLUING
α and then one can use the infinite dimensional implicit function in the space
of all maps w with eR (w) ∈ α to find a nearby J-holomorphic curve w̃j
with eR (w̃j ) ∈ α. Of course, this produces the same J-holomorphic curves
intersecting α, however some additional argument is required to show that
the intersection numbers are the same for both invariants ΨA (α1 , α2 , α3 , α4 )
and ΨA−B,B (α1 , α2 ; α3 , α4 ).
(ii) The techniques of this section can obviously be generalized to glue together
finitely many intersecting J-holomorphic curves described by a framed class
D = (A1 , . . . , AN , j2 , . . . , jN ).
(iii) The map fR constructed above will in general not preserve any group action. The best way to deal with this problem seems to be to use additional
intersection conditions as gauge fixing as in (i) above.
(iv) The above techniques can also be used to glue together J-holomorphic maps
ui : Σi → M which are defined on Riemann surfaces of higher genus. However,
in this case we must allow for the complex structure on the glued Riemann
surface Σ = Σ1 #Σ2 to vary. The gluing will produce a Riemann surface with
a very thin neck. Conversely, it follows from Gromov compactness that a
sequence uν : Σ → M of (jν , J)-holomorphic curves can only split up into two
(or more) surfaces of higher genus if the complex structure jν on Σ converges
to the boundary of Teichmüller space.
Appendix B
Elliptic Regularity
B.1
Sobolev spaces
Throughout Ω ⊂ Rn is an open set with smooth boundary. Denote by C ∞ (Ω̄) the
space of restrictions of smooth functions on Rn to Ω̄ and by C0∞ (Ω) the space of
smooth compactly supported functions on Ω.
For a positive integer k and a number 1 ≤ p < ∞ define the W k,p -norm of a
smooth function u : Ω → R by
kukk,p

Z
=
1/p
X
|∂ ν u(x)|p dx
Ω |ν|≤k
where ν = (ν1 , . . . , νn ) is a multi-index and |ν| = ν1 + · · · + νn . The Sobolev space
W k,p (Ω)
is defined as the completion of C ∞ (Ω̄) with respect to the W k,p -norm. The space
W0k,p (Ω)
is the closure of C0∞ (Ω) in W k,p (Ω).
The next proposition uses the smoothness of the boundary of Ω to show that
every function with weak Lp -derivatives up to order k lies in the space W k,p (Ω).
Proposition B.1.1 Let u ∈ Lp (Ω). Then u ∈ W k,p (Ω) if and only if for every
|ν| ≤ k there exists a function uν ∈ Lp (Ω) such that
Z
Z
ν
|ν|
u(x)∂ φ(x) dx = (−1)
uν (x)φ(x) dx
Ω
Ω
for φ ∈ C0∞ (Ω).
In this case we define the weak derivative ∂ ν u to be uν .
181
182
APPENDIX B. ELLIPTIC REGULARITY
Proof: We sketch the main idea in the case Ω = {x ∈ Rn | x1 > 0}. Assume that
u has weak Lp derivatives up to order k. Let ρ : Rn → R be a smooth nonnegative
function such that
Z
supp ρ ⊂ B1 ,
ρ(x) dx = 1.
Rn
Define
ρδ (x) = δ −n ρ(δ −1 x).
Now choose a smooth cutoff function β : Rn → [0, 1] such that β(x) = 1 for |x| < 1
and β(x) = 0 for |x| > 2. Then the functions
uδ (x) = β(δx)u(x1 + δ, x2 , . . . , xn )
have compact support, are defined for x1 > −δ, and have weak Lp -derivatives up to
order k. These converge in the Lp -norm to the weak derivatives of u. The functions
Z
ρδ ∗ uδ (x) =
ρδ (x − y)uδ (y) dy
Rn
are smooth with compact support and converge in the W k,p -norm. The limit is u.
Hence u ∈ W k,p (Ω). The general case can be reduced to the above by choosing
suitable local coordinates near every boundary point.
2
It is somewhat less than obvious that a function u ∈ W 1,p (Ω) whose derivatives
all vanish must be constant on every component of Ω. The proof requires the
following fundamental estimate. As always, ∇u denotes the gradient of the function
u. Recall that the mean value of a function is its integral over its domain of
definition divided by the volume of the domain.
Lemma B.1.2 (Poincaré’s inequality) Let 1 < p < ∞ and Ω ⊂ Rn be a
bounded open domain. Then there exists a constant c = c(p, n, Ω) > 0 such that
kukLp (Ω) ≤ c k∇ukLp (Ω)
for every u ∈ C0∞ (Ω). If Ω is a square then this continues to hold for u ∈ C ∞ (Ω̄)
with mean value zero.
Proof: The first statement is an easy exercise. The second statement on functions with mean value zero is proved by induction over n. Assume without loss of
generality that Ω is the unit square Qn = {x ∈ Rn | 0 < xj < 1}. For n = 1 the
statement is again an easy exercise. Assume that the statement is proved for n ≥ 1
and let u ∈ C ∞ (Qn+1 ) be of mean value zero. Define v ∈ C ∞ (Q1 ) by
Z
v(t) =
u(x1 , . . . , xn , t) dx1 · · · dxn .
Qn
Since v is of mean value 0
Z
Z 1
p
|v(t)| dt ≤ c1
0
0
1
p
Z
|v̇(t)| dt ≤ c1
Q
∂u p
dx.
n+1 ∂xn+1
B.1. SOBOLEV SPACES
183
The last step follows from Hölder’s inequality. By the induction hypothesis
Z
Z
p
|u(x, t) − v(t)| dx ≤ c2
|∇u(x, t)|p dx.
Qn
Qn
Integrate over t and use the previous inequality to obtain the required estimate. 2
Corollary B.1.3 Let Ω ⊂ Rn be a bounded open domain and u ∈ W 1,p (Ω) with
weak derivatives ∂u/∂xj ≡ 0 for j = 1, . . . , n. Then u is constant on each connected
component of Ω. If, moreover, u ∈ W01,p (Ω) then u ≡ 0.
Proof: First assume that Ω is a square and u has mean value zero. Approximate u
in the W 1,p -norm by a sequence of smooth functions uν ∈ C ∞ (Ω̄) with mean value
zero. Then Poincaré’s inequality shows that uν converges to zero in the Lp -norm.
Hence u = 0. This shows u is constant on every square where its first derivatives
vanish. Hence a function u ∈ W 1,p (Ω) with ∇u ≡ 0 is constant on every connected
component of Ω. The same argument works for u ∈ W01,p (Ω).
2
A function with weak derivatives need not be continuous. Consider for example
the function
u(x) = |x|−α
with α ∈ R in the domain Ω = B1 = {x ∈ Rn | |x| < 1}. Then
∂u
= −αxj |x|−α−2 .
∂xj
By induction,
|∂ ν u(x)| ≤ cν |x|−α−|ν| .
Now the function x 7→ |x|−β is integrable on B1 if and only if β < n. Hence the
derivatives of u up to order k will be p-integrable whenever
αp + kp < n.
If kp < n choose 0 < α < n/p − k to obtain a function which is in W k,p (B1 ) but
not continuous at 0. For kp > n this construction fails and, in fact, in this case
every W k,p -function is continuous. More precisely, define the Hölder norm
|u(x) − u(y)|
+ sup |u(x)|
|x − y|ε
x,y∈Ω
x∈Ω
kukC ε = sup
for 0 < ε ≤ 1 and
kukC k+ε =
X
k∂ ν ukC ε .
|ν|≤k
k+ε
Denote by C
(Ω) the space of all C k -functions u : Ω → R with finite Hölder
norm kukC k+ε .
Theorem B.1.4 Assume kp > n. Then there exists a constant c = c(k, p, Ω) > 0
such that
n
kukC ε ≤ c kukW k,p ,
ε=k− .
p
In particular, by the Arzela-Ascoli theorem, the injection W k,p (Ω) ,→ C 0 (Ω) is
compact.
184
Theorem B.1.5 Assume kp < n. Then there exists a constant c = c(k, p, Ω) > 0
such that
np
kukLq ≤ c kukW k,p ,
q=
.
n − kp
If q < np/(n − kp) then the injection W k,p (Ω) ,→ Lq (Ω) is compact.
These are the Sobolev estimates. The compactness statement in Theorem B.1.5 is known as Rellich’s theorem. The case kp = n is the borderline
situation for these estimates. In this case the space W k,p does not embed into
the space of continuous functions. Of particular interest here is the case where
n = p = 2 and k = 1.
Remark B.1.6 In Lemma A.1.1 we have seen that there exists a sequence of function uj ∈ W 1,2 (B1 ) on the unit disc B1 = {(x, y) ∈ R2 | x2 + y 2 < 1} such that
uj (0) = 1,
lim kuj kW 1,2 = 0.
j→∞
In 2 dimensions the relation between W 1,2 and C 0 is rather subtle. For example the
function u(eiθ ) = θ, 0 ≤ θ < 2π, (with a single jump discontinuity) does not extend
to a W 1,2 -function an B1 . On the other hand there exist discontinuous functions
on S 1 which do extend to W 1,2 -functions on B1 .
2
The case kp > n should be viewed as the good case where everything works. For
example composition with a smooth function and products.
Proposition B.1.7 Assume kp > n. Then there exists a constant c = c(k, p) > 0
such that
kuvkW k,p ≤ c (kukW k,p kvkL∞ + kukL∞ kvkW k,p )
kf ◦ ukW k,p ≤ c (kf kC k + 1) kukW k,p
∞
for u, v ∈ C (Ω) and f ∈ Ck (R).
Proposition B.1.8 Assume kp > n and f ∈ C ∞ (R). Then the map
W k,p (Ω) → W k,p (Ω) : u 7→ f ◦ u
is a C ∞ -map of Banach spaces.
Remark B.1.9 Let M be an n-dimensional smooth compact manifold and π :
E → M be a smooth vector bundle. A section s : M → E is said to be of
class W k,p if all its local coordinate representations are in W k,p . This definition is
independent of the choice of the coordinates. To see this note that if φ ∈ Diff(Rn ),
k,p
Φ ∈ C ∞ (Rn , RN ×N ) is matrix-valued function, and u ∈ Wloc
(Rn , RN ) then Φ(u ◦
k,p
n
φ) ∈ W (Ω) for every bounded open set Ω ⊂ R and there is an estimate
kΦ(u ◦ φ)kW k,p (Ω) ≤ c kukW k,p (Ω)
with c = c(φ, Φ, Ω) independent of u. This holds even when kp ≤ n. For functions
with values in a manifold the situation is quite different. Proposition B.1.7 shows
that for such functions it is required that kp > n. To define a norm on the space
of W k,p -sections one can take the sum of the W k,p -norms over finitely many charts
which cover M .
2
B.2. THE CALDERON-ZYGMUND INEQUALITY
B.2
185
The Calderon-Zygmund inequality
Denote by
∂2
∂2
∂2
+
+
·
·
·
+
∂x1 2
∂x2 2
∂xn 2
n
2
the Laplace-operator on R . A C -function u : Ω → R on an open set Ω ⊂ Rn
is called harmonic if ∆u = 0. Harmonic functions are real analytic. (If n = 2
then a function is harmonic iff it is locally the real part of a holomorphic function.)
Harmonic functions are characterized by the mean value property (see John [33])
Z
n
u(ξ) dξ,
Br (x) ⊂ Ω.
u(x) =
ωn r2 Br (x)
∆=
Here ωn = 2π n/2 Γ(n/2)−1 is the volume of the unit sphere in Rn . In particular,
ω2 = 2π.
The fundamental solution of Laplace’s equation is
(2π)−1 log |x|,
n = 2,
K(x) =
n ≥ 3.
(2 − n)−1 ωn−1 |x|2−n ,
Its first and second derivatives are given by
Kj (x) =
xj
,
ωn |x|n
Kjk (x) =
−nxj xk
,
ωn |x|n+2
Kjj (x) =
|x|2 − nx2j
ωn |x|n+2
where Kj = ∂K/∂xj and Kjk = ∂ 2 K/∂xj ∂xk . In particular, ∆K = 0. The
function K and its first derivatives Kj are integrable on compact sets while the
second derivatives are not. Hence ∂j (K ∗ f ) = Kj ∗ f for compactly supported
functions f : Rn → R but there is no such formula for the second derivatives.
Moreover, since neither K nor its derivatives are integrable on Rn , care must be
taken for functions f which do not have compact support.
Every compactly supported C 2 -function u : Rn → R satisfies
u = K ∗ ∆u,
∂j u = Kj ∗ ∆u.
where ∗ denotes convolution. Conversely,
∆(K ∗ f ) = f,
∆(Kj ∗ f ) = ∂j f
for f ∈ C0∞ (Rn ) (see [33]). This is Poisson’s identity. In general K ∗ f will
not have compact support. Since the second derivatives of K are not integrable on
compact sets there exists a continuous function f : Rn → R such that K ∗ f ∈
/ C 2.
For such a function f there is no classical solution of ∆u = f . (This follows from
the next two lemmata.) The situation is however quite different for weak solutions.
Let f ∈ L1loc (Ω). A function u ∈ L1loc (Ω) is called a weak solution of ∆u = f if
Z
Z
u(x)∆φ(x) dx =
f (x)φ(x) dx
Ω
Ω
for φ ∈ C0∞ (Ω). Similarly u ∈ L1loc (Ω) is called a weak solution of ∆u = ∂j f with
f ∈ L1loc if
Z
Z
u(x)∆φ(x) dx = −
f (x)∂j φ(x) dx
Ω
for φ ∈ C0∞ (Ω).
Ω
186
Lemma B.2.1 Let u, f ∈ L1 (Rn ) with compact support.
(i) u is a weak solution of ∆u = f if and only if u = K ∗ f .
(i) u is a weak solution of ∆u = ∂j f if and only if u = Kj ∗ f .
Proof: If u = K ∗ f . Then
Z
Z
Z
Z
u∆φ = (K ∗ f )∆φ = f (K ∗ ∆φ) = f φ
for φ ∈ C0∞ (Rn ). Conversely, suppose that u is a weak solution of ∆u = f . Choose
ρδ : Rn → R as in the proof of Proposition B.1.1. Then
Z
Z
Z
Z
(∆ρδ ∗ u)φ = u(∆ρδ ∗ φ) = f (ρδ ∗ φ) = (ρδ ∗ f )φ.
for every φ ∈ C0∞ (Rn ) and hence ∆(ρδ ∗ u) = (∆ρδ ) ∗ u = ρδ ∗ f . This implies that
ρδ ∗ u − K ∗ ρδ ∗ f is a bounded harmonic function converging to zero at ∞ and
hence ρδ ∗ u = K ∗ ρδ ∗ f. Take the limit δ → 0 to obtain u = K ∗ f . This proves (i).
The proof of (ii) is similar and is left to the reader.
2
Lemma B.2.2 (Weyl’s lemma) Every weak solution u ∈ L1loc (Ω) of ∆u = 0 is
harmonic.
Proof: Let ρδ be as in the proof of Lemma B.2.1. The function uδ = ρδ ∗ u is
harmonic in Ωδ = {x ∈ Ω | Bδ (x) ⊂ Ω}. Hence uδ satisfies the mean value property.
Since uδ converges to u in the L1 -norm on every compact subset of Ω it follows
that u has the mean value property. Hence u is harmonic (cf. [33]).
2
Theorem B.2.3 (Calderon-Zygmund inequality) For 1 < p < ∞ there exists
a constant c = c(n, p) > 0 such that
k∇(Kj ∗ f )kLp ≤ c kf kLp
(B.1)
for f ∈ C0∞ (Rn ) and j = 1, . . . , n.
This theorem is the fundamental estimate for the Lp -theory of elliptic operators.
We include here a proof which is a modification of the one given by Gilbarg and
Trudinger [23]. The proof requires the following four lemmata. The first is the case
p = 2.
Lemma B.2.4 The estimate (B.1) holds for p = 2.
Proof: Since u(x) = Kj ∗ f (x) converges to zero as |x| tends to infinity we have
2
k∇ukL2 = h∇u, ∇ui = −hu, ∆ui = −hu, ∂j f i = h∂j u, f i ≤ k∇ukL2 kf kL2 .
Here all inner products are L2 -inner products on Rn . Divide both sides by k∇ukL2
to obtain the required estimate.
2
For any measurable function f : Rn → R define
µ(t, f ) = {x ∈ R2 | |f (x)| > t}
for t > 0 where |A| denotes the Lebesgues measure of the set A.
187
Lemma B.2.5 For 1 ≤ p < ∞ and f ∈ Lp (Rn )
Z
Z ∞
tp µ(t, f ) ≤ |f (x)|p dx = p
sp−1 µ(s, f ) ds.
0
Moreover,
µ(t, f + g) ≤ µ(t/2, f ) + µ(t/2, g).
Proof: Integrate the function F : Rn+1 → R defined by F (x, t) = ptp−1 for
0 ≤ t ≤ |f (x)| and F (x, t) = 0 otherwise.
2
Apply the previous Lemma to the function ∂k (Kj ∗ f ). By Lemma B.2.4
k∂k (Kj ∗ f )kL2 ≤ kf kL2
and hence
µ(t, ∂k (Kj ∗ f )) ≤
1
t2
Z
|f (x)|2 dx
(B.2)
The next two lemmata establish a similar inequality with the L2 -norm on the right
replaced by the L1 -norm. Theorem B.2.3 is proved by interpolation for 1 < p < 2.
Lemma B.2.6 There exists a constant c = c(n) > 0 such that every function
f ∈ L1 (Rn ) with supp f ⊂ Br and mean value 0 satisfies
1
µ(t, ∂k (Kj ∗ f )) ≤ c rn + kf kL1
t
for j, k = 1, . . . , n.
Proof: Denote ukj = ∂k (Kj ∗ f ). For |x| > r
Z
ukj (x) =
(∂k Kj (x − y) − ∂k Kj (x)) f (y) dy
Br
≤
max |∂k Kj (x − y) − ∂k Kj (x)| kf kL1
y∈Br
≤ r max |∇∂k Kj (x − y)| kf kL1
y∈Br
≤ r max
y∈Br
≤
n3/2 (n + 2)
kf kL1
ωn |x − y|n+1
n3/2 (n + 2)r
kf kL1 .
ωn (|x| − r)n+1
Hence
Z
|ukj (x)| dx
≤
|x|>2r
=
n3/2 (n + 2)r
ωn
n
3/2
(n + 2)r
ωn
Z
dx
kf kL1
(|x| − r)n+1
|x|>2r
∞
ωn ρn−1 dρ
n+1
2r (ρ − r)
Z ∞
Z
≤ 2n−1 n3/2 (n + 2)r
2r
=
kf kL1
dρ
kf kL1
(ρ − r)2
2n−2 n3/2 (n + 2) kf kL1 .
188
This implies
t|B2r | + |{x ∈ Rn | |ukj (x)| > t, |x| > 2r}|
Z
≤ t|B2r | +
|ukj (x)| dx
tµ(t, ukj ) ≤
|x|>2r
=
tωn 2n rn
+ 2n−2 n3/2 (n + 2) kf kL1 .
n
2
Lemma B.2.7 There exists a constant c = c(n) > 0 such that the characteristic
function f = χQ of any square Q ⊂ Rn satisfies
Z
c
µ(t, ∂k (Kj ∗ f )) ≤
|f (x)| dx
t
for j, k = 1, . . . , n.
Proof: Assume without loss of generality that Q = Qr = {x ∈ Rn | |xj | < r}. For
t > 1 the left hand side is zero. Hence assume 0 < t ≤ 1 and write
f = f0 + f1 ,
Then
f0 = tχQR ,
Z
Z
f0 =
f1 = χQr − tχQR ,
f = (2r)n ,
R = t−1/n r.
supp f1 ⊂ B√nR ,
and f1 is of mean value zero. Let c1 > 0 be the constant of Lemma B.2.6. Then
n/2 n
1
c2
n r
µ(t, ∂k (Kj ∗ f1 )) ≤ c1
+ kf1 kL1 ≤
kf kL1 .
t
t
t
where c2 = c1 (nn/2 2−n + 2). By Lemma B.2.5 and Lemma B.2.4
µ(t, ∂k (Kj ∗ f0 )) ≤
1
1
1
1
k∂k (Kj ∗ f0 )kL2 ≤ 2 kf0 kL2 = kf0 kL1 = kf kL1 .
2
t
t
t
t
Again by Lemma B.2.5
µ(t, ∂k (Kj ∗ f )) ≤ µ(t/2, ∂k (Kj ∗ f0 )) + µ(t/2, ∂k (Kj ∗ f1 )) ≤
2 + 2c2
kf kL1 . 2
t
The previous estimate holds for all f ∈ L2 ∩ L1 [23]. However, for the proof of
Theorem B.2.3 it is only needed in the case where f is a step function.
Proof of Theorem B.2.3: First assume 1 < p < 2. It suffices to prove the
estimate for characteristic functions of squares. Hence assume f = χQ for some
open square Q ⊂ Rn . Then by Lemma B.2.5
Z
Z ∞
p
|∂k (Kj ∗ f )(x)| dx = p
tp−1 µ(t, ∂k (Kj ∗ f )) dt
Rn
0
Z 1
Z ∞
p−2
p−3
≤
pc
t
dt + p
t
dt kf kL1
0
1
pc
p
=
+
kf kLp .
p−1 2−p
189
Here we have used Lemma B.2.7 for t < 1 and (B.2) for t > 1. This proves the
estimate for 1 < p < 2. For 2 < p < ∞ we use duality. Let 1 < q < 2 such that
1/p + 1/q = 1. Then
Z
Z
g(x)∂k (Kj ∗ f )(x) dx = ∂k (Kj ∗ g)(x)f (x) ≤ c kf kLp kgkLq
2
and hence k∂k (Kj ∗ f )kLp ≤ c kf kLp .
Theorem B.2.8 (Local regularity) Let 1 < p < ∞, k ≥ 0 be an integer, and
Ω ⊂ Rn be an open domain. If u ∈ Lploc (Ω) is a weak solution of ∆u = f with
k,p
k+2,p
f ∈ Wloc
(Ω) then u ∈ Wloc
(Ω). Moreover, for every compact set Q ⊂ Ω there
exists a constant c = c(k, p, n, Q, Ω) > 0 such that
kukW k+2,p (Q) ≤ c k∆ukW k,p (Ω) + kukLp (Ω)
for u ∈ C ∞ (Ω̄).
Proof: Let u and f be as in (i) and Q as in (ii). Choose an open neighborhood U
of Q such that cl(U ) ⊂ Ω. Let β ∈ C0∞ (Ω) be a smooth cutoff function such that
β(x) = 1 for x ∈ U and define
v = K ∗ βf.
k+2,p
It follows from Theorem B.2.3 that v ∈ Wloc
(Ω). By Lemma B.2.1 v is a weak
solution of ∆v = βf . Hence the restriction of u − v to U is a weak solution of
∆(u − v) = 0. By Weyl’s lemma u − v is real analytic in U . Hence u ∈ W k+2,p (Q).
This proves the first statement.
By Theorem B.2.3, the function v = K ∗ β∆u satisfies an estimate
kvkW k+2,p (U ) ≤ c1 kβ∆ukW k,p (Ω) ≤ c2 k∆ukW k,p (Ω) .
The function v − u is harmonic in U . By the mean value property for harmonic
functions there exists a constant c3 > 0 such that
kv − ukW k+2,p (Q) ≤ c3 kv − ukLp (U ) ≤ c3 kvkW k+2,p (U ) + kukLp (U ) .
Take the sum of these inequalities to obtain the required estimate.
2
Sometimes it is useful to consider weak solutions of ∆u = f where f is not a
function but a distribution in W −1,p . We rephrase this in terms of weak solutions
of ∆u = divf with f ∈ Lp .
Theorem B.2.9 Let 1 < p < ∞, k ≥ 0 be an integer, and Ω ⊂ Rn be an open
domain. Assume that u ∈ Lploc (Ω) and f = (f0 , . . . , fn ) ∈ Lploc (Ω, Rn+1 ) satisfy
Z
Z
n Z
X
u(x)∆φ(x) dx =
f0 (x)φ(x) dx −
fj (x)∂j φ(x) dx
Ω
Ω
j=1
Ω
1,p
(Ω) and for every compact set Q ⊂ Ω there
for every φ ∈ C0∞ (Ω). Then u ∈ Wloc
is an estimate
kukW 1,p (Q) ≤ c kf kLp (Ω) + kukLp (Ω)
where c = c(p, n, Q, Ω) > 0 is independent of u.
190
Proof: Choose an open neighbourhood U of Q with cl(U ) ⊂ Ω. Let β ∈ C0∞ (Ω)
be a smooth cutoff function such that β(x) = 1 for x ∈ U . Define
v = K ∗ βf0 +
n
X
Kj ∗ βfj .
j=1
1,p
It follows from Theorem B.2.3 that v ∈ Wloc
(Ω) and there an estimate
kvkW 1,p (U ) ≤ c1 kβf kLp (Ω) ≤ c1 kf kLp (Ω) .
Here we have also used Poincaré’s inequality. Now, by Lemma B.2.1, v is a weak
solution of
n
X
∆v = βf0 +
∂j (βfj )
j=1
where ∂j (βfj ) is to be understood as a distribution. Hence the restriction of u − v
to Q is a weak solution of ∆(u − v) = 0. By Weyl’s lemma u − v is harmonic in U .
Hence u ∈ W 1,p (Q). Moreover, by the mean value property for harmonic functions,
there exists a constant c2 > 0 such that
kv − ukW 1,p (Q) ≤ c2 kv − ukLp (U ) ≤ c2 kvkW 1,p (U ) + kukLp (U ) .
Take the sum of these inequalities to obtain the reqired estimate.
B.3
2
Cauchy-Riemann operators
Identify R2 = C with z = x + iy. For z = x + iy and ζ = ξ + iη denote
hz, ζi = xξ + yη.
In this section all functions take values in C. For example C0∞ (Ω) denotes the
space of smooth compactly supported functions Ω → C. Consider the first order
differential operators
1 ∂
∂
∂
1 ∂
∂
∂
=
+i
,
=
−i
.
∂ z̄
2 ∂x
∂y
∂z
2 ∂x
∂y
A C 1 -function u : Ω → C on an open set Ω ⊂ C satisfies the Cauchy-Riemann
equation ∂u/∂ z̄ = 0 if and only if it is holomorphic. It satisfies ∂u/∂z = 0 iff it is
anti-holomorphic. The following lemma shows that the function
N (z) =
1
.
πz
is the fundamental solution of the Cauchy-Riemann operator ∂/∂ z̄.
Lemma B.3.1 If f : C → C is a smooth compactly supported function then u =
N ∗ f is a solution of the inhomogeneous Cauchy Riemann equations ∂u/∂ z̄ = f .
B.3. CAUCHY-RIEMANN OPERATORS
191
Proof: Note that
∆=4
∂ ∂
,
∂z ∂ z̄
N =4
∂K
,
∂z
K(z) =
log |z|
.
2π
Hence
∂
∂
(N ∗ f ) = 4
∂ z̄
∂ z̄
∂K
∗f
∂z
=4
∂ ∂
(K ∗ f ) = ∆(K ∗ f ) = f
∂ z̄ ∂z
2
for f ∈ C0∞ (C)
By the divergence theorem
Z
Z
h∂v/∂z, ui + hv, ∂u/∂ z̄i = 0
Ω
Ω
for u, v ∈ C0∞ (Ω). In other words −∂/∂z is the formal adjoint operator of ∂/∂ z̄. A
function u ∈ L1loc (Ω) is called a weak solution of ∂u/∂ z̄ = f for f ∈ L1loc (Ω) if
Z
Z
h∂φ/∂z, ui +
Ω
hφ, f i = 0
(B.3)
Ω
for φ ∈ C0∞ (Ω).
Lemma B.3.2 Let u, f ∈ Lp (C) with compact support. Then u is a weak solution
of ∂u/∂ z̄ = f if and only if u = N ∗ f .
Proof: Assume first u = N ∗ f . Then
Z
Z
Z
Z
hu, ∂φ/∂zi = hN ∗ f, ∂φ/∂zi = − hf, N̄ ∗ ∂φ/∂zi = − hf, φi
Ω
Ω
Ω
Ω
for φ ∈ C0∞ (Ω). Hence u is a weak solution of ∂u/∂ z̄ = f . Conversely, suppose
that u satisfies (B.3) and choose ρδ : C → R as in the proof of Proposition B.1.1
with n = 2. Then
∂
ρδ ∗ u = ρδ ∗ f.
∂ z̄
Since ρδ ∗ u ∈ C0∞ (C) it follows from by Lemma B.3.1 that ρδ ∗ u − N ∗ ρδ ∗ f is a
bounded holomorphic function converging to zero at ∞ and hence ρδ ∗u = N ∗ρδ ∗f .
Take the limit δ → 0 to obtain u = N ∗ f .
2
Lemma B.3.3 Every weak solution u ∈ L1loc (Ω) of ∂u/∂ z̄ = 0 is holomorphic.
Proof: If u is a weak soluion of ∂u/∂ z̄ = 0 then its real and imaginary part are
weak solutions of Laplace’s equation. By Weyl’s lemma they are smooth. Hence u
is holomorphic.
2
192
Theorem B.3.4 (Local regularity) Let 1 < p < ∞, k ≥ 0 be an integer, and
Ω ⊂ C be an open domain. If u ∈ Lploc (Ω) is a weak solution of ∂u/∂ z̄ = f with
k,p
k+1,p
f ∈ Wloc
(Ω) then u ∈ Wloc
(Ω). Moreover, for every compact set Q ⊂ Ω there
exists a constant c = c(k, p, Q, Ω) > 0 such that
!
∂u + kukLp (Ω)
kukW k+1,p (Q) ≤ c ∂ z̄ k,p
W
(Ω)
for u ∈ C ∞ (Ω̄).
Proof: Let u and f be as in (i) and Q as in (ii). Choose a compact neighborhood
U of Q such that U ⊂ Ω. Let β ∈ C0∞ (Ω) be a smooth cutoff function such that
β(x) = 1 for x ∈ U and define
v = N ∗ βf.
k+1,p
By Theorem B.2.3 v ∈ Wloc
(Ω). By Lemma B.3.2 v is a weak solution of
∂v/∂ z̄ = βf . Hence the restriction of u − v to U is a weak solution of
∂(u − v)
= 0.
∂ z̄
By Lemma B.3.3 u − v is holomorphic in int(U ). Hence u ∈ W k+1,p (Q). This
proves the first statement. The proof of the estimate is as in Theorem B.2.8 and is
left to the reader.
2
Exercise B.3.5 Use the previous theorem to prove the estimate (3.3) of Chapter 3.
Use it also to prove that the cokernel of Du is the kernel of Du∗ , or more precisely,
if η ∈ Lq (Λ0,1 T ∗ Σ ⊗J u∗ T M ) satisfies hη, Du ξi = 0 for all ξ ∈ W 1,p (u∗ T M ) then
η ∈ W 1,q and Du∗ η = 0. Hint: In a suitable trivialization u∗ T M the operator Du
is of the form ∂/∂ z̄ + zero-th order term. Use the formula of Remark 3.3.3.
2
B.4
Elliptic bootstrapping
In this section we shall prove the following two theorems about the smoothness of
J-holomorphic curves and compactness for sequences with uniform bounds on the
Lp -norm of the derivatives with p > 2. Assume that (M, J) is an almost complex
manifold. Let Σ be an oriented Riemann surface without boundary with complex
structure j.
Theorem B.4.1 (Regularity) If u : Σ → M is a J-holomorphic curve of class
W k,p with kp > 2 then u is smooth.
Theorem B.4.2 (Compactness) Let Jν be a sequence of almost complex structures on M converging to J in the C ∞ -topology and jν be a sequence of complex
structures on Σ converging to j in the C ∞ -topology. Let Uν ⊂ Σ be an increasing sequence of open sets whose union is Σ and uν : Uν → M be a sequence of
(jν , Jν )-holomorphic curves. Assume that for every compact set Q ⊂ Σ there exists
a compact set K ⊂ M and constants p > 2 and c > 0 such that
kduν kLp (Q) ≤ c,
uν (Q) ⊂ K
B.4. ELLIPTIC BOOTSTRAPPING
193
for ν sufficiently large. Then a subsequence of the uν converges uniformly with all
derivatives on compact sets to a (j, J)-holomorphic curve u : Σ → M .
Both these theorems are obvious when J is integrable and Σ is closed because
each component of a J-holomorphic curve in holomorphic coordinates is a harmonic
function. In particular, the compactness theorem follows from the mean value
property of harmonic functions. The general case is considerably harder.
We begin by proving a local estimate for the solutions of linear Cauchy-Riemann
equations. Let Ω ⊂ C be an open set and u : Ω → R2n be a solution of the linear
PDE
∂s u + J∂t u = η,
(B.4)
where J : Ω → R2n×2n with J 2 = −1l and η : Ω → R2n . We shall first prove
regularity of u under the weakest possible regularity assumptions on the almost
complex structure J and the function η. Eventually we shall consider the semilinear
case where J is replaced by a function of the form J ◦ u. Here J is a smooth almost
complex structure on R2n .
k,p
Denote by Wloc
(Ω, R2n ) the set of those functions u : Ω → R2n such that the
restriction of u to any compact subset of Ω is of class W k,p . We shall first assume
1,p
that the almost complex structure J is of class Wloc
. If u is a solution of (B.4)
then, by partial integration,
Z
Z
h∂s φ + J T ∂t φ, ui dsdt = − hφ, η + (∂t J)ui dsdt
(B.5)
Ω
Ω
for every test function φ ∈ C0∞ (Ω, R2n ). (Here J T denotes the transpose of J.) The
next lemma asserts the converse.
pq/(p+q)
1,p
Lemma B.4.3 Assume J ∈ Wloc
and η ∈ Lloc
where p > 2 and p/(p − 1) <
1,pq/(p+q)
q
∞
q ≤ ∞. If u ∈ Lloc satisfies (B.5) for every φ ∈ C0 then u ∈ Wloc
and u
satisfies (B.4) almost everywhere. Moreover, for every compact set Q ⊂ Ω there is
an estimate
kukW 1,pq/(p+q) (Q) ≤ c kukLq (Ω) + kηkLpq/(p+q) (Ω)
where the constant c = c(p, q, Q, Ω, kJkW 1,p ) > 0 is independent of u.
1
Proof: Let ψ : Ω → R2n be a smooth test function with compact support and
define φ = ∂s ψ − J T ∂t ψ. Then
∂s φ + J T ∂t φ = ∆ψ − (∂s J)T ∂t ψ + (∂t J)T J T ∂t ψ
and hence (B.5) implies
Z
Z
h∆ψ, ui = −
Ω
Z
h∂s ψ, f i +
Ω
h∂t ψ, gi.
(B.6)
g = (∂s J)u + Jη.
(B.7)
Ω
where
f = (∂t J)u + η,
1,p
1 In the case q = ∞ we have η ∈ Lp
loc and u ∈ Wloc and the estimate holds with the corresponding norms.
194
pq/(p+q)
By assumption and Hölder’s inequality the functions f and g are of class Lloc
Equation B.6 asserts that u is a weak solution of
.
∆u = ∂s f − ∂t g
where the right hand side is to be understood in the distributional sense. By
1,pq/(p+q)
Theorem B.2.9 we have u ∈ Wloc
and for every compact set Q ⊂ Ω there is
an estimate
kukW 1,pq/(p+q) (Q) ≤ c1 kf kLpq/(p+q) (Ω) + kgkLpq/(p+q) (Ω) + kukLpq/(p+q) (Ω)
≤ c1 kJkW 1,p (Ω) kukLq (Ω) + kukLpq/(p+q) (Ω)
+ kJkL∞ (Ω) kηkLpq/(p+q) (Ω) + kηkLpq/(p+q) (Ω)
≤ c2 kukLq (Ω) + kηkLpq/(p+q) (Ω) .
Here the constants c1 and c2 depend on p, q, Q, Ω, and the W 1,p -norm of J but not
on u. If follows from integration by parts that u satisfies (B.4) almost everywhere.
2
Lemma B.4.4 Given p > 2 there exists a finite increasing sequence q0 < q1 <
· · · < q` such that
p
< q0 ≤ p,
p−1
and
qj+1 =
2rj
,
2 − rj
q`−1 <
rj =
2p
< q` ,
p−2
pqj
,
p + qj
for j = 0, . . . , ` − 1.
Proof: Consider the map h : (p/(p − 1), 2p/(p − 2)) → (2, ∞) defined by
h(q) =
2r
2pq
=
,
2p + 2q − pq
2−r
r=
pq
< 2.
p+q
The condition r < 2 is equivalent to q < 2p/(p − 2). The map h is a monotonically
increasing diffeomorphism such that h(q) > q. Now choose the sequence qj such
that qj+1 = h(qj ).
2
Lemma B.4.5 Assume p > 2 and 1 < r ≤ p. If f ∈ W 1,p and g ∈ W 1,r then
f g ∈ W 1,r with
kf gkW 1,r ≤ c kf kW 1,p kgkW 1,r .
Proof: Examine the Lr -norm of the expression expression d(f g) = (df )g + f (dg).
The term f (dg) can be estimated by the sup-norm of f and the W 1,r -norm of g.
The term (df )g can be estimated by
k(df )gkLr ≤ kdf kLp kgkLq ,
1
1 1
+ = .
p q
r
If r < 2 then, since p > 2, we have q = pr/(p − r) < 2r/(2 − r) and the Lq norm of
g can be estimated by the W 1,r -norm. The latter is obvious when r ≥ 2.
2
B.4. ELLIPTIC BOOTSTRAPPING
195
1,p
Lemma B.4.6 Assume J ∈ Wloc
and η ∈ Lploc where p > 2. If u ∈ Lploc satis1,p
∞
fies (B.5) for every φ ∈ C0 then u ∈ Wloc
. Moreover, for every compact set Q ⊂ Ω
there is an estimate
kukW 1,p (Q) ≤ c kukLp (Ω) + kηkLp (Ω)
with c = c(p, Q, Ω, kJkW 1,p ) > 0 independent of u.
Proof: By Lemma B.4.3 and the Sobolev embedding theorem we have
u ∈ Lqloc ,
p
2p
<q<
p−1
p−2
=⇒
0
u ∈ W 1,r ⊂ Lqloc ,
where r = pq/(p + q) < 2 and q 0 = 2r/(2 − r) > q. Now choose qj and rj as in
1,r
Lemma B.4.4. By induction we have u ∈ Wloc j for every j and
kukW 1,rj (Qj ) ≤ c1 kukLqj (Qj−1 ) + kηkLrj (Qj−1 )
≤ c2 kukW 1,rj−1 (Qj−1 ) + kηkLp (Ω)
≤ c3 kukW 1,r0 (Q0 ) + kηkLp (Ω)
≤ c4 kukLp (Ω) + kηkLp (Ω)
where Qj+1 ⊂ int(Qj ) and Q ⊂ int(Q` ). In the first and last inequality we have
used Lemma B.4.3. In the last inequality we have also used q0 ≤ p. The second
inequality follows from the Sobolev embedding theorem and the third by induction.
With j = ` we have r` = pq` /(p + q` ) > 2 and hence u is continuous. Now use
Lemma B.4.3 again with q = ∞.
2
k,p
k,p
Proposition B.4.7 Assume J ∈ Wloc
and η ∈ Wloc
where p > 2 and k ≥ 1. If
p
k+1,p
∞
u ∈ Lloc satisfies (B.5) for every φ ∈ C0 then u ∈ Wloc
. Moreover, for every
compact set Q ⊂ Ω there is an estimate
kukW k+1,p (Q) ≤ c kukW k,p (Ω) + kηkW k,p (Ω)
with c = c(p, Q, Ω, kJkW k,p ) > 0 independent of u.
Proof: First assume k = 1. Then
1,q
u ∈ Wloc
,
p
<q≤∞
p−1
2,pq/(p+q)
u ∈ Wloc
=⇒
and u satisfies the obvious estimate. To see this note that the function
u0 = ∂s u ∈ Lqloc
is a weak solution of (B.4) with η replaced by
pq/(p+q)
η 0 = ∂s η − (∂s J)∂t u ∈ Lloc
.
(B.8)
196
1,pq/(p+q)
Hence it follows from Lemma B.4.3 that u0 ∈ Wloc
. By Lemma B.4.5 ∂t u =
1,pq/(p+q)
2,pq/(p+q)
J(∂s u − η) ∈ Wloc
and hence u ∈ Wloc
.
To prove the statement for k = 1 choose sequences qj and rj as in Lemma B.4.4
2,r
and use (B.8) inductively as in the proof of Lemma B.4.6 to obtain u ∈ Wloc j for
every j. With j = ` it follows that u is continuously differentiable and, by (B.8)
2,p
with q = ∞, we have u ∈ Wloc
. This proves the assertion for k = 1. For general
k it is proved by induction. Assume the proposition is true for k ≥ 1. Suppose
k+1,p
k+1,p
that J ∈ Wloc
and η ∈ Wloc
. Apply the induction hypothesis to u0 and η 0 as
k+1,p
above to obtain that ∂s u = u0 and ∂t u = J(∂s u − η) are of class Wloc
. Hence
k+2,p
u ∈ Wloc
and this proves the proposition with k replaced by k + 1.
2
Proof of Theorem B.4.1: It suffices to prove the theorem in local coordinates.
Hence we shall assume that Ω ⊂ C is an open set and u : Ω → R2n is a solution of
the PDE
∂u
∂u
+ J(u)
= 0,
∂s
∂t
where J : R2n → R2n×2n is a smooth almost complex structure. Now assume first
1,p
that k = 1. Then u ∈ Wloc
(Ω, R2n ) satisfies the requirements of Proposition B.4.7
1,p
2,p
with k = 1, η = 0, and J replaced by J ◦ u ∈ Wloc
. Hence u ∈ Wloc
and
2,p
k,p
J ◦ u ∈ Wloc . Continue by induction to obtain that u ∈ Wloc for every k. Hence u
k,p
is smooth. This proves the theorem for k = 1. If u ∈ Wloc
with (k − 1)p ≥ 2 then
1,q
u ∈ Wloc
for every q and the statement follows from the case k = 1. If (k − 1)p < 2
1,q
then u ∈ Wloc
with q = 2p/(2 − kp + p) > 2 and the statement follows again from
the case k = 1.
2
Proof of Theorem B.4.2: Since the inclusion W 1,p ,→ C is compact for p > 2
we may assume that uν converges uniformly to a continuous function u : Σ → M .
Now, in local coordinates, we may assume that uν : Ω → R2n is a sequence of
smooth solutions of the PDE
∂s uν + Jν (uν )∂t uν = 0,
where Jν converges to J in the C ∞ -topology. It follows from Proposition B.4.7 by
induction that
sup kukW k,p (Q) < ∞
ν
for every k and every compact subset Q ⊂ Ω. By the Arzela-Ascoli theorem the
inclusion W k,p ,→ C k−1 is compact. This proves the existence of a subsequence
which converges in the C k−1 -topology on every compact subset of Σ. The theorem
follows by choosing a diagonal subsequence.
2
Exercise B.4.8 Find the simplest proof you can of Theorems B.4.1 and B.4.2
when J is integrable and Σ is closed.
2
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Index
a priori estimate
for energy density, 44
action
Morse-Novikov theory for, 154
of short loop, 45
on loop space, 153
adjunction formula, 39, 101
almost complex structure
K-positive, 59
K-semi-positive, 59
ω-compatible, 25
ω-tame, 2, 42
condition to be regular, 38
generic, 5
good, 98
integrable, 3
positive, 59
regular, 24
semi-positive, 59, 143
smooth homotopy, 25
tame versus compatible, 26, 60
Arnold conjecture, 158, 165
Aronszajn, 15, 197
Aronszajn’s theorem, 14, 35
Aspinwall, ix, 107, 151, 197
Aspinwall-Morrison formula, 151
Astashkevich, 119, 197
Atiyah, 197
Atiyah-Floer conjecture, 163, 165
Audin, ix, 1, 197
Bertram, 123, 197
bubbling, 41, 46–49, 155
Calabi-Yau manifold, 1, 12, 137, 141,
145, 148–151
examples, 148, 149
Calderon-Zygmund inequality, 186–189
Callaghan, 163, 197
Candelas, 141, 148, 197
Cauchy-Riemann equation, 2, 5, 13–15,
190
perturbed, 118, 144, 193
Cauchy-Riemann operator, 24
fundamental solution, 190
Chern character, 128
Chern class, 2
quantum, 121
Chern number, 38
minimal, 10, 61
negative, 86, 165
zero, 148
Ciocan-Fontanine, 119, 197
cohomology
quantum, 108
compactness, 1, 5–6
compactness theorem, 51, 63, 65, 68
proof, 52, 81–87
composition rule, 137
condition
(H1), 93, 101
(H2), 93, 96
(H3), 96
(H4), 101, 103, 114
(H5), 101, 103, 114
(JA3 ), 67
(JA4 ), 68, 86, 142
(JAp ), 67–69, 85–87, 101
conformal map, 13, 41
conformal rescaling, 41
Conley, 155
convergence problem, 137, 139, 141, 149,
151
critical point, 15
finite number of, 15
stable and unstable manifolds, 161
cross-ratio, 105
cup product
203
204
deformed, 110, 147
weakly monotone case, 141
given by triple intersection, 107
pair-of-pants, 160
curve, 3–4
approximate J-holomorphic, 30
as symplectic submanifold, 3
counting discrete curves, 101, 138,
151
critical point, 15–18
determined by ∞-jet, 14
energy, 6
graph of, 142
implicit function theorem for, 30
injective point, 4, 18
intersections of, 17
isolated, 89
multiply-covered, 4, 18, 119, 142–
144, 165
of negative Chern number, 86, 165
parametrized, 3
positivity of intersections, 21, 39, 100,
101
reducible, 42
regular, 24, 38
simple, 4, 18–21
singular point of sequence, 53
somewhere injective, 4, 18
is simple, 18
weak convergence, 50
cusp-curve, 42, 50, 62
framing D, 63, 76
label T , 65, 81
moduli space of, 75
type D, 76
with two components, 80
Du
definition, 24
ellipticity of, 28
formula for, 28
its adjoint Du∗ , 29
right inverse, 29, 173–176
surjectivity of, 29, 38
Darboux’s theorem, 2
Daskalopoulos, 123, 197
determinant bundle, 33
dimension condition
INDEX
for Φ, 93
for Ψ, 102
for a ∗ b, 110
for a ? b ? c, 114
Donaldson, 33, 162, 167, 168, 197
Donaldson’s quantum category, 162–165
for Lagrangians, 164
for mapping tori, 163
Dostoglou, 30, 155, 162, 163, 197
Dubrovin, 107, 113, 131, 139, 198
Dubrovin connection, 113, 132
and quantum products, 133
explicit formula, 134
flatness, 136
potential function, 135
elliptic bootstrapping, 41, 43, 192–196
elliptic regularity, 25–27, 29, 35, 191, 192
energy, 6, 42, 44
bounds derivative, 44
conformal invariance, 41
identity, 41, 43
energy level, 145
evaluation map, 6–8, 49
and orientations, 98
as pseudo-cycle, 94–98
domain of, 64
for cusp-curves, 79–81
for marked curves, 66–69
image of, 62–65
is submersion, 71–74
p-fold, 8, 65, 66
compactification, 8, 65
finiteness condition, 144, 147, 156
finiteness result, 155
first Chern class, 2
Floer, ix, 12, 15, 41, 153, 155–158, 162,
164, 198
Floer (co)homology, 1, 11, 12, 153–159
and quantum cohomology, 160
cochain complex, 155
Euler characteristic, 164
of (H, J), 156
of symplectomorphism, 162
pair-of-pants product, 160
ring structure, 159–160
Floer-Donaldson theory, 163
INDEX
framing
D for cusp-curve, 63, 76
Fredholm operator, 5
determinant bundle, 33
index, 5, 24
regular value, 5, 33
Fredholm theory, 1, 4, 5
Frobenius algebra, 113, 131
Frobenius condition, 133
Gilbarg, 186, 188, 198
Givental, ix, 11, 107, 119, 121, 198
Green, 141, 148, 197
Griffiths, 33, 38, 100, 198
Gromov, ix, 7, 41, 50, 89, 107, 143, 198
Gromov invariant Φ, 66, 89, 93–94
dimension condition, 93
for 6-manifolds, 100
for conics in CP 2 , 99
for discrete curves, 101
for lines in CP n , 98
on blow up, 100
Gromov’s compactness theorem, 50–58
Gromov-Witten invariant Ψ, 66, 69, 101–
105
Gromov-Witten invariants, 1, 8–9
compared on CP 2 , 104
composition rule, 117
on CP 2 , 119
deformation invariance, 96
mixed, 90, 138
signs, 98
well defined for Kähler manifolds,
96
Gromov-Witten potential, 131–139
for CP 2 , 138
for CP n , 138
non-monotone case, 139
Grothendieck, 38, 198
Guillemin, 92, 198
Hölder inequality, 169
Hölder norm, 183
Hamiltonian differential equation, 153
harmonic function, 185
mean value property, 185
205
Harris, 33, 38, 100, 198
Hartman, 15, 198
Heegard splitting, 165
Hofer, 12, 15, 40, 41, 51, 54, 146, 153,
155, 157, 158, 162, 198
homology class
J-effective, 67
framed, 63
indecomposable, 49
of pure degree, 108, 131
spherical, 6, 47
Hurewicz homomorphism, 6
implicit function theorem, 27–33
integrable systems, 113, 121
isoperimetric inequality, 41, 45
(J, J 0 )-holomorphic, 2
J-holomorphic map, 13
John, 185, 186, 199
Kähler manifold, 61
and symplectic deformations, 100
Fano variety, 60
Gromov-Witten invariants, 96
Kim, 11, 107, 119, 121, 198
Kobayashi, 199
Kobayaski, 28
Kodaira vanishing theorem, 38
Kontsevich, 90, 113, 131, 137–139, 199
Kronheimer, 167, 168, 197
labelling
T for cusp-curve, 65, 81
Lafontaine, ix, 1, 197
Lalonde, ix, 7, 199
Landau-Ginzburg potential, 126, 127
Laplace’s equation
fundamental solution, 185
Laurent polynomial ring, 10
Lawson, 44, 199
Liu, 118, 199
Lizan, 40
loop
action of, 45
loop space
of manifold, 146, 154
universal cover, 154
Lorek, 40, 98, 199
206
Manin, 90, 113, 131, 137–139, 199
mapping tori
and Atiyah-Floer conjecture, 163
marked sphere, 105
Maslov index, 155
mass
of singular point, 53
McDuff, ix, 1, 2, 7, 17, 21, 28, 33, 39, 40,
59, 62, 74, 77, 89, 98–101, 156,
199, 200
Micallef, 21, 200
Milnor, 92, 129, 200
minimal Chern number, 10, 61
mirror symmetry conjecture, 122, 148,
149
moduli space, 4–5
cobordism of, 25
compactifications, 59–69
complex structure on, 33
integration over, 97, 103
main theorems, 23–25
of N -tuples of curves, 74–75
of cusp-curves, 75–78
of flat connections, 163
of two-component cusp-curves, 80
of unparametrized curves, 52
orientation, 32, 33, 98
regular point of, 24
top strata in the boundary, 82
universal, 33, 77
Yang-Mills, 33
Morrison, ix, 107, 151, 197
Morse-Novikov theory, 146, 154
Morse-Witten coboundary, 158
Morse-Witten complex , 161
Moser, 121, 122, 200
Nijenhuis, 74, 200
Nijenhuis tensor, 28
anti-commutes with J, 32
Nomizu, 28, 199
non-squeezing theorem, 7
Novikov, 146, 154, 200
Novikov ring, 11, 12, 141, 144–148
as Laurent series ring, 146
number of curves
in CP 2 , 105, 138
in Calabi-Yau manifold, 101, 151
INDEX
Oh, 200
Ono, 153, 158, 200
Ossa, 141, 148, 197
Pansu, 41, 44, 51, 200
Parker, 41, 51, 200
Parkes, 141, 148, 197
Piunikhin, 157, 161, 200
Poisson’s identity, 185
Pollack, 92, 198
potential function, 135
pseudo-cycle, 8, 63, 66, 68, 90–93
bordant, 90
of dimension k, 90
transverse, 91
weak representative, 93
quantum
category, 162, 164
Chern classes, 121
quantum cohomology, 1, 9–11, 108–110
and Floer cohomology, 157, 160
as Lagrangian variety, 122
cup product, 110–114, 147
associativity, 114–119, 161
Frobenius structure of, 113, 131
of CP n , 11, 113
of flag manifold, 11, 110, 119–122,
146
of Grassmannian, 123–130
periodic form, 12, 147
with Novikov rings, 146
rational maps Ratm of CP 1 , 83, 150
regular
J-holomorphic curve, 24, 39
value of Fredholm operator, 33
Rellich’s theorem, 27, 184
removal of singularities, 41, 43–46
reparametrization group G, 42, 62
Riemann-Roch theorem, 24, 33
Ruan, ix, 9, 33, 50, 59, 62, 64, 66, 76,
89, 90, 100, 101, 104, 105, 107,
117, 118, 131, 137, 138, 144,
200, 201
Sacks, 41, 201
INDEX
Sadov, 119, 161, 197, 201
Salamon, ix, 1, 2, 7, 12, 15, 30, 41, 44,
51, 100, 146, 153, 155–158, 161–
163, 197–201
Sard’s theorem, 5, 24
Sard-Smale theorem, 33, 36, 78
Schwarz, 157, 158, 161, 201
semi-positivity, 59
Siebert, 121, 123, 124, 201
sigma model, 89
sign
convention for Lie bracket, 132
of intersection, 98
Sikorav, 40
singular point of curve, 53
Smale, 33, 201
smoothness
of class W k,p , 181, 184
smoothness convention, 34
Sobolev embedding theorem, 27, 184
Sobolev estimate
borderline case, 6, 41, 184
Sobolev space, 181–184
spherical, 47
Stasheff, 129, 200
supermanifold, 131, 139
symplectic form
deformation, 96
symplectic manifold, 1–3
monotone, 1, 8, 60
non-equivalent, 100
weakly monotone, 59–61, 141, 153
Arnold conjecture for, 158
Taubes, ix, 33, 36, 201
Tian, ix, 90, 104, 107, 117, 118, 121, 123,
124, 131, 137, 138, 144, 201
Toda lattice, 119, 121
triple intersection product, 107
Trudinger, 186, 188, 198
Uhlenbeck, 41, 201
unique continuation, 15
universal moduli space, 33, 77
is a manifold, 34, 77
Vafa, ix, 107, 201
Verlinde algebra, 123, 128–130
207
gluing rules, 130
Viterbo, 160, 201
WDVV-equation, 11, 135, 137
weak convergence
definition, 50
weak derivative, 181
weak solution, 185, 191
Wentworth, 123, 197
Weyl’s lemma, 186
White, 21, 200
Wintner, 15, 198
Witten, 89, 107, 123, 124, 130, 158, 202
Wolfson, 41, 51, 200
Woolf, 74, 200
Yau, 148
Ye, 41, 51, 202
Zehnder, 155–157, 201
208
INDEX
Index of Notations
Jτ (M, ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
End(T M, J, ω) . . . . . . . . . . . . . . . . . . . . .34
c1 (A). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
π : M` (A, J ) → J ` . . . . . . . . . . . . . . . . 36
u : (Σ, j) → (M, J) . . . . . . . . . . . . . . . . . . 3
gJ (v, w) = hv, wiJ . . . . . . . . . . . . . . . . . . 42
C = Im u . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
E(u; Br ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
M(A, J) . . . . . . . . . . . . . . . . . . . . . . . . 4, 23
C = C 1 ∪ C 2 ∪ · · · ∪ C N . . . . . . . . . . . . 50
Jreg (A) . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 24
u = (u1 , . . . , uN ) . . . . . . . . . . . . . . . . . . . 50
hv, wi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
A(r, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
G = PSL(2, C) . . . . . . . . . . . . . . . . . . . . . . 6
J+ (M, ω), J+ (M, ω, K) . . . . . . . . . . . . 59
W(A, J) = M(A, J) ×G CP 1 . . . . 7, 62
D = {A1 , . . . , AN , j2 , . . . , jN } . . . . . . . 63
eJ : W(A, J) → M . . . . . . . . . . . . . . . . . . 7
W(D, J). . . . . . . . . . . . . . . . . . . . . . . .64, 79
ΦA (α1 , . . . , αp ) . . . . . . . . . . . . . . . . . . 9, 94
ep : W(A, J, p) → M p . . . . . . . . . . . . . . .65
ΨA (α1 , . . . , αp ) . . . . . . . . . . . . . . . . . .9, 99
z = (z1 , . . . , zp ) . . . . . . . . . . . . . . . . . . . . . 66
α = PD(a) . . . . . . . . . . . . . . . . . . . .10, 106
ez = eA,J,z : M(A, J) → M p . . . . . . . . 66
a ∗ b . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 108
eD,T,z : V(D, T, J, z) → M p . . . . . . . . . 68
QH ∗ (M ) . . . . . . . . . . . . . . . . . . . . . 10, 106
M(A1 , . . . , AN , J ) . . . . . . . . . . . . . . . . . 74
˜ ∗ (M ) = H ∗ (M ) ⊗ Z[q] . . . . . 10, 107
QH
∆N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Γ = Im π2 (M ) → H2 (M ) . . . . . . 11, 142
πD : M(D, J ) → J . . . . . . . . . . . . . . . . 78
Λω . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 142
ΦA,p : Hd (M p , Z) → Z. . . . . . . . . . . . . .93
∂¯J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
ΨA,p : Hd (M p , Z) → Z . . . . . . . . . . . . 102
X = Map(Σ, M ; A) . . . . . . . . . . . . . . . . 23
ha, bi . . . . . . . . . . . . . . . . . . . . . . . . . 108, 131
Du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24, 28
(a ∗ b)A . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Jreg (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
ΨA,B (α, β; γ, δ) . . . . . . . . . . . . . . . . . . . 116
J (M, ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
b A
b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
J,
(u, J) ∈ M` (A, J ) ⊂ X k,p × J ` . . . . 34
W k,p (Ω), W0k,p (Ω) . . . . . . . . . . . . . . . . 181
209
210
INDEX

J-holomorphic Curves and Quantum Cohomology

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